GC26-3847-0
File No. S370-22
Systems
APL Language
FT 5 }
PREFACE
This manual is intended to provide a reference for the APL language,
and also an introduction to APL to those unfamiliar with it. New users
should read the introduction, which provides an overview of some
fundamental concepts, and illustrates several kinds of use.
Section 2 presents the fundamentals and introduces the terminology
used in later sections, but the sections may otherwise be read
relcitively independently. For example, a glance at the listing of the
primitive functions in Figures 3.1 and 3.5 should suffice for the reader
who wishes to concentrate just on the means of defining and using
functions (programs) presented in Sections 6 and 7.
Readers familiar with earlier versions of APL but not with the
notions of shared variables, will find this topic and the related topics
of system functions and system variables covered in Sections 4 and 5.
The following publications are intended to supplement the information
in this publication for particular APL systems:
APL Shared Variables (APLSV) User's Guide, Order No. SH20-1460
APL/CMS User's Guide, Order No. SC20-1846
First Edition (March 1975)
This publication serves two different APL systems: APL Shared
Variables (Program Number 5799-AJF — Programming RPQ WE1191) and
APL/CMS (Program Number 5799-ALK — Programming RPQ MF2608) .
Changes may from time to time be made to the specifications herein; any
such changes will be reported in subsequent revisions or Technical
Newsletters. Before using this publication, consult the latest IBM
System/360 and System/370 Bibliography , Order No. GA22-6822, for the
editions that are applicable and current.
Reqxiests for copies of IBM publications should be made to your IBM
representative, or to the IBM branch office serving your locality.
A form for readers ' comments is provided at the back of this
publication. If the form has been removed, comments may be addressed to
IBM Corporation
Systems Development Division
LDF Publishing — Department JO 4
1501 California Avenue
Palo Alto, CA 94304.
©Copyright International Business Machines Corporation 1975
CONTENTS
SECTION 1 : INTRODUCTION ..... 1
Two Examples of the Use of APL 1
An Isolated Calculation 1
A Prepared Workspace 2
The Characteristics of APL 7
SECTION 2: FUNDAMENTALS 9
Character Set 10
Scalar and Vector Constants ... 10
Spaces 13
Functions 13
Order of Execution 14
Arrays 14
Workspaces and Libraries 15
Names 16
User Identification 16
SECTION 3: PRIMITIVE FUNCTIONS AND OPERATORS 17
Scalar Functions 17
Plus, Minus, Times, Divide and Residue 19
Conjugate, Negative, Signum, Reciprocal and Magnitude 20
Boolean and Relational Functions 20
Minimum and Maximum 21
Floor and Ceiling 21
Roll (Random Number Function) 22
Power, Exponential, General and Natural Logarithm 22
Circular, Hyperbolic, and Pythagorean Functions 23
Factorial and Binomial Functions 24
Operators 25
Reduction 25
Scan 26
Axis 27
Inner Product 27
Outer Product '. 29
Mixed Functions 29
Structural Functions 31
Shape, Reshape, and Ravel 31
Reverse and Rotate 33
Catenate and Laminate 33
Transpose 35
Selection Functions 36
Take and Drop 36
Compress and Expand 36
Indexing 37
Selector Generators 38
Index Generator and Index of 38
Membership 39
Grade Functions 39
Deal 39
Numerical Functions . 40
Matrix Inverse and Matrix Divide 40
Decode and Encode 42
Data Transformations 43
Execute and Format 43
SECTION 4: SYSTEM FUNCTIONS AND SYSTEM VARIABLES 47
System Functions 47
Canonical Representation 47
Function Establishment .... 47
Expunge 48
Name List 49
Name Classification 49
Delay 49
System Variables 50
Contents iii
SECTION 5: SHARED VARIABLES 52
Offers „ 52
Access Control . 55
Retraction 58
Inquiries 58
SECTION 6: FRICTION DEFINITION 59
Canonical Representation and Function Establishment ....... 59
The Function Header 60
Local and. Global Names 61
Branchincr and Statement Numbers 61
Labels 62
Comments 62
Function Ed.itina - the V Form 6 3
Addincr a Statement 63
Inserting or Replacina a Statement 6 3
Replacing the Header 63
Deleting a Statement 6 3
Editing a Statement or Header 6 4
Addina to a Statement or Header 6 4
Function Display 6 4
Leaving the V Form 65
SECTION 7: FRICTION EXECUTION 66
Halted Execution 66
State Indicator ......... 67
State Indicator Damage 68
Trace Control . 68
Stop Control 68
Locked Functions 69
Recursive Functions 69
Terminal Input and Output 70
Evaluated Input 71
Interrupting Evaluated Input ... 71
Character Input 71
Interrupting Character Input .... 71
Normal Output 72
Bare Output . 72
SECTION 8: SYSTEM COMMANDS .... 73
Commands that Modify the Workspace Environment .... 76
Grouping to Facilitate Copyina or Erasure .... 78
Commands that Monitor the Active Workspace .... 79
Commands for Workspace Storage and Retrieval ... 80
Libraries of Saved Workspaces .... 80
Names and Passwords for Workspaces 80
Automatic Savina after Line Drop
•and Special Properties of the C OUT I HUE Workspace ....... 83
Commands Recrarding Workspace Storage and Retrieval ........ 83
Access to the System 83
Sign On 84
Sign Off 84
INDEX 87
iv APL Language
ILLUSTRATIONS
Figure
Title
Paae
Figure
1.
,1
Figure
2.
,1
Figure
2.
.2
Figure
2,
.3
Figure
3,
.1
Figure
3.
,2
Fiaure
3.
,3
Figure
3.
,4
Figure
3.
,5
Figure
3.
,6
Figure
3.
,7
Figure
4,
.1
Figure
4.
,2
Figure
5,
.1
Figure
5,
.2
Figure
5,
,3
Figure
8,
.1
Figure
8,
,2
Figure
8.
.3
Functions for Sales, Billina, and Inventory Control ... 6
Error Reports 11
The APL Character Set 12
Typical APL Keyboard ......... 12
Primitive Scalar Functions 18
Identity Elements of Primitive Scalar Dyadic Functions . 19
The Pythagorean Functions 2 4
Inner Product 28
Primitive fixed Functions 30
Scalar-Vector Substitutions for fixed Functions .... 31
Shape and Rank Vectors 32
System Functions 48
System Variables 51
Functions for the Management of Sharing 5 3
Access Control of a Shared Variable 56
Some Useful Settings for the Access Control Vector ... 57
System Commands 7 4
Trouble Reports 75
Environment Within a Clear Workspace 76
Illustrations
SECTION 1 ; INTRODUCTION
APL is a general-purpose language which enjoys extensive use in
such diverse applications as commercial data processing, system design,
mathematical and scientific computation, and the teaching of mathematics
and other subjects. It has proved particularly useful in data-base
applications, where its computational power and communication facilities
combine to enhance the productivity of both application programmers and
end users.
When implemented as a computing system, APL is used from a
typewriter-like keyboard. Statements that specify the work to be done
are entered by typing them, and in response the computer displays the
result of the computation. The result appears at a device which
accompanies the keyboard, such as a printer, or video display. In
addition to work purely at the keyboard and its associated display,
entries may also invoke the use of printers, disk-files, tapes or other
remote devices .
TWO EXAMPLES OF THE USE OF APL
A statement entered at the keyboard may contain numbers , or
symbols such as + - x *, or names formed from letters of the alphabet.
The numbers and special symbols stand for the primitive objects and
functions of APL; primitive in the sense that their meanings are
permanently fixed, and therefore understood by the APL system without
further definition. A name, however, has no significance until a
meaning has been assigned to it.
Names are used for two major categories of objects. There are
names for collections of data, made up of numbers or characters. Such a
named collection is. called a variable . Names may also be used for
programs made up of sequences of APL statements. Such programs are
called defined functions . Once they have been established, names of
variables and defined functions can be used in statements by themselves
or in combination with the primitive functions and objects.
AN ISOLATED CALCULATION
If the work to be done can be adequately specified simply by
keying a statement made up of numbers and symbols, names will not be
required: entering the expression to be evaluated causes the result to
be displayed. For example, suppose it is required to compare the rates
of return on money at a fixed interest rate but with different
compounding intervals. For 1000 units at 6%, compounded annually,
quarterly, monthly, or daily, for 10 years, the transaction (at a
typewriter terminal) would look like this:
1000x(l+. 06*1 4 12 365)*10xl 4 12 365 (entry)
1790.85 1814.02 1819.4 1822.03 (response)
(The largest gain is apparently obtained in going from annually to
quarterly, and after that the differences are relatively insignificant.)
Several distinctive features of APL are illustrated in this
example: familiar symbols such as + x * are used where possible;
symbols are introduced where necessary, (as the * for the power
function) ; and a group of numbers can be worked on together.
Section 1 : Introduction
A PREPARED WORKSPACE
Although there are many problems that can be solved by keying in
the appropriate numbers and symbols, the greatest benefits of using APL
come when named functions and data are used. Since a single name may
refer to a large array of data, using the name is far simpler than
keying in all of its members. Similarly, a defined function, invoked by
entering its name , may be composed of many individual APL statements
that, would be burdensome to type individually again and again.
Once a function has been defined, or data collected under a name,
it is usually desirable to retain the significance of the names for
some period of time; perhaps for just a few minutes, but more often for
much longer periods, possibly months or years. For this reason APL
systems are organized around the concept of a workspace , which might be
thought of as a notebook in which all the different items needed during
some piece of work are recorded together.
The following example illustrates the use of an application
package - a workspace prepared for a particular data-processing
application. It is not expected that the reader new to APL will
immediately follow all details, but the example should nevertheless
provide, on the one hand, an indication of the ease of exploring and
using such a package, and on the other hand, the relative ease of
constructing it.
The example is a skeletal system for sales, billing, and inventory
control for, say, a wholesale distributor of gadgets. The system
maintains an inventory of items in stock, with part numbers,
descriptions, unit prices, quantities on hand, and reorder levels. It
automatically adjusts the quantity on hand when an invoice is prepared,
and signals when the stock level goes below the reorder level. It also
provides for entering new stock items in the inventory list and has
provision for order entry and printing of invoices.
To complete this basic system for actual application, it would be
necessary to add functions such as the recording of a journal and the
maintenance of the customer list, and to add appropriate checks on
errors of input and on security. Nevertheless, the package does work,
as far as it goes, and in addition to showing the use of many functions
of the language, also demonstrates the conciseness of APL programming.
To start, the stored workspace containing the application package
is activated, using a system command :
)LOAD 1 SBIC
SAVED 18.42.25 02/06/75
To become acquainted with the contents of the package the names of
variables and defined functions are listed:
)VARS
CUSTLIST DESCRIBE INVNO LOW MTH ORDLIST
STOCKLIST
)FNS
ADDRESS DATE GET INVOICE NEWSTOCK ORDER ORDERENTRI
PRINT PUT RESTOCK
APL Language
The variable name DESCRIBE is suggestive. Its value can be
displayed by simply entering the name:
DESCRIBE
THIS WORKSPACE CONTAINS FUNCTIONS FOR RECORDING ORDERS,
MAINTAINING AN INVENTORY, AND PREPARING INVOICES. IT IS A
SKELETAL SYSTEM DESIGNED TO ILLUSTRATE THE USE OF APL IN
COMMERCIAL DATA PROCESSING.
THE VARIABLES CUSTLIST , ORDLIST , AND STOCKLIST CARRY,
RESPECTIVELY , CUSTOMER NAMES AND ADDRESSES KEYED TO A
CUSTOMER NUMBER, ORDERS BY STOCK NUMBER AND CUSTOMER NUMBER,
AND THE INVENTORY LIST OF STOCK ITEMS. INVNO IS A COUNTER
FOR SUCCESSIVE INVOICE NUMBERS, LOW INDICATES ITEMS TO BE
REORDERED, AND MTH HOLDS NAMES FOR MONTHS OF THE YEAR.
WHEN DISPLAYING THE STOCK LIST OR CUSTOMER LIST ON A
TYPEWRITER TERMINAL IT WILL BE DESIRABLE TO CHANGE TO A
TYPE-ELEMENT HAVING BOTH UPPER AND LOWER CASE ALPHABETS.
MEANINGFUL STATEMENTS IN THIS WORKSPACE ARE:
NEWSTOCK
ORDERENTRY
PRINT INVOICE ORDER N
RESTOCK
WHERE N IS A CUSTOMER NUMBER. FOR CONSISTENCY WITH THE DATA
ALREADY ENTERED, NEWSTOCK AND PRINT SHOULD ALSO BE USED
WITH A TYPE-ELEMENT THAT HAS BOTH ALPHABETS.
THE FUNCTIONS ADDRESS, DATE, GET, AND PUT ARE NOT
NORMALLY INVOKED BY A DIRECT KEYBOARD ENTRY, BUT ARE USED BY
THE VARIOUS FUNCTIONS NOTED ABOVE.
It might be helpful at this point to display STOCKLIST to see just
what the information looks like. But the list might be very large, and
it is good practice to find out its size before asking for the display:
pSTOCKLIST
6 39
The list has 6 rows and 39 columns, which is not too much to
display in its entirety. Following the advice in DESCRIBE the
type-element can be changed and the variable name entered:
stocklist
1135 First Great Item
9993 High Flyer Widget
3569 Second Moneymaker
5613 Mail Order Special
2583 A Real Winner
9998 Nonesuch Frammis
Next it might be useful to look at the customer list:
pCUSTLIST
3 5 20
9.95
350
55
88.73
240
35
24.75
200
30
14.99
600
95
49.99
125
10
2.69
500
50
Section 1 : Introduction
The customer list is apparently represented by a three-dimensional
array, and it will be interesting to see how it is displayed:
custlist
7
City Traders Inc.
41 Postage Road
Rimela, N. Y. 12345
55
Mail House Ltd.
7-11 Ramblers Lane
Offshore island
City, S. Dak. 54321
312
Mantup Sales Corp.
Ruralia Farms
RFD 2
Suburbian, Wis. 0000
Finally, it is probably worth looking at the order list:
pORDLIST
6 3
ORDLIST
55 3569 5
312 9998 12
7 1135 2
312 3569 10
312 5613 45
7 5613 75
This is a 3-column numerical array, and the significance of
each column is easily deduced from the previous displays: the sucessive
columns appear to represent customer number, stock number, and quantity.
Use of the package can be demonstrated by trying the meaningful
statements :
ORDERENTRY
CUSTOMER ITEM QUANTITY
In response to the statement ORDERENTRY the system has displayed column
headings, and the keyboard has unlocked for further entry. These can be
supplied, using some of the existing customer and item numbers. The
entire transaction at the terminal might be this :
ORDERENTRY
CUSTOMER ITEM QUANTITY
55 1135 12
312 1135 24
55 2583 4
Since customer 5 5 had one item on order before, three should be on order
now.
APL Language
Although it is not listed as a meaningful statement, what happens
if the function ORDER is used by itself?
ORDER 5 5
55 3569 5
55 1135 12
55 2583 4
This entrv produces a result which seems to confirm the number of items
on "rderf It turns out that this result will be used by INVOICE to
produce a result which will in turn be used by PRINT.
invoice order 55
1336
3569 Second Moneymaker
1 1 35 First Great Item
2583 A Real Winner
If the complete statement given in DESCRIBE is entered rather then
these piecemeal entries, the result is a formatted output xntended to be
used with a pre-printed form:
print invoice order 55
55
24.75
5
123.75
9.95
12
119.40
49.99
4
199.96
To:
Mail House Ltd.
7-11 Ramblers Lane
Offshore Island
City, S. Dak. 54321
Invoice No.
1336
Feb 8, 1975
Stock
3569
1135
2583
Description
Second Moneymaker
First Great Item
A Real Winner
Price
24.75
9.95
49.99
Quan.
5
12
4
Exten.
123.75
119.40
199.96
Total $ 443.11
Distributors Corp. 123 First Ave., City, St. 99999
For the sake of completeness, and because they illustrate
variations in the method of requesting input, the remaining two
statements can be entered. NEWSTOCK prints a prompting message and then
unlocks the keyboard, waiting for entry of the response on the same
line:
newstock
stock number/ 8760
description/ The Best Gadget
unit price/ 1.73
initial inventory/ 875
reorder level/ 100
RESTOCK not only prints a prompting message, but uses an
additional prompt D: supplied by the system:
RESTOCK
STOCK NUMBER
D:
1135
STOCK INCREMENT
D:
30
Section 1 : Introduction
9
95
368
55
88
73
240
35
24
75
195
30
14
99
600
95
49
99
121
10
2.
69
500
50
It might now be of interest to display the inventory once more to
see the effect of the previous operations:
stocklist
1135 First Great Item
9993 High Flyer Widget
3569 Second Moneymaker
5613 Mail Order Special
2583 A Real Winner
999 8 Nonesuch Frammis
8760 The Best Gadget 1.73 875 100
The quantities on hand do indeed show the effect of completing the
order, replenishing the stock, and adding a new item.
The complete set of ten functions in this package is displayed in
Figure 1.1. The first line in each definition is the function head er,
which specifies, among other things, the function name and syntax"! — The
remainder in each individual display is the function body . The longest
function, INVOICE, has twelve lines, the shortest, ORDER, has only one.
FORM-SPRINT N;T;X
X+l+pN
F0RM<r~H<b 12 43 +ADDRESS 1 ~6 +W
F0flM[l;37+i6>6+ff[l; ]
FORMl3;31+\12]+DATE QTS
F0RMi5+xX-l;l«- 1 +ff
F0RML12; ]-<-~43 + T + /i , 1 35 IN
1])
Z+INVOICE N;R;X;PQR;EXT
X+ 6 VlNVNO+INVNO+1
Z+ 1 43 p(X, 37 TtfCl;
PQR+&.251R+GET J[2]
EXT-rXL3lxPQRH]
2*Z,[l3(31+i?), 4 8 2 ?Xl3l,EXT
PQRL21+PQR121-X131
■+(>/PQRL2 3])/£2
L0W+L0WA~Xl2leL0W) /XZ21
£2:fl[31+i4]-e 4 fPQi?[2]
R PUT ATC 2]
-»-(0*l + pilM- 1 +ff)/Ll
Z+ORDER N
Z«-( ORDLISTl -,U=N) /ORDLIST
Z+ADDRESS Nil
J> ( CUSTLISTl ;l;i6]A. = ,tf)il
Z«- 1 ICUSTLISTH; ;]
Z+DATE N
Z+-MTHlNL2l;l, 3 6 «[3 1]
Z[7>»,»
NEWSTOCKiR
EH-' STOCK NUMBER/ •
R+ 4 Tid
CD-*-' DESCRIPTION/ '
i?-«-fl,21 + 12 + [Il
ek ' wit price/ '
i?+-i?, 6 2 TIB
PH-'JffJ2\ZVl£ INVENTORY/
R+R, 4 t±H
EH-' REORDER LEVEL/ '
fl«-fl, 4 TiE
i? PUTsl^*R
ORDERENTRI ; L
' CUSTOMER ITEM QUANTITY*
LI :-►( ' 'A. =L-f-[3)/0 <** -
ORDLIST+ORDLIST,tlli.L
■+L1
RESTOCK;R;X;PQR
< STOCK NUMBER'
PQR<r±25iR+GET X<-Q
' STOCK INCREMENT'
i?[31 + i4]«- 4 VPQRL21+PQR121+Q
R PUT X
R PUT Nil
I+{STOCKLIST\_ ;i4]a.= 4 wN)\l
+ (.I*l + pSTOCKLIST)/Ll
STOCKLIST+STOCKLIST. Ill ' '
L1-.ST0CKLISTLI; >i?
Z-<-0£T #;I
If-(52 , 0CXLJ52'C;i4]A.= 4 Ti7) 1 1
Z+STOCKLISTHil
Figure 1.1: Functions for Sales, Billing, and Inventory Control
APL Language
The details of the programming will not be discussed here, but one
point is worth noting. As they stand here the various data lists are
very small. In a real use, one or more of them might have many
thousands of entries. A good way to handle them in that case is
illustrated by the treatment of STOCKLIST, which is accessed only by the
functions put and GET, rather than the primitive APL functions which
could have been used directly for this purpose. By thus isolating the
access functions the application package design is automatically made
more general; if a different representation were chosen for STOCKLIST -
say, an external file- the required changes in the programming would be
confined to the definitions of PUT and GET, and the other functions such
as INVOICE or NEWSTOCK, which give the package its particular character,
would be unaffected. In this way, .an APL application can be expanded to
data-bases of arbitrary size, well beyond the storage capacity of the
workspace that establishes the environment for the work.
THE CHARACTERISTICS OF APL
The subsequent sections of this manual describe APL in detail,
giving the meaning of each symbol and discussing the various features
common to the APL systems currently available from IBM. It is well to
view these, details in light of the major characteristics of APL, which
may be summarized as follows:
The primitive objects of the language are arrays (lists, tables,
lists of tables, etc.). For example, A+B is meaningful for any
arrays A and B, the size of an array \pA) is a primitive function,
and arrays may be indexed by arrays, as in yl[3 l 4- 2]
The syntax is simple: there are only three statement types (name
assignment, branch, or neither) , there is no function precedence
hierarchy, functions have either one, two, or no arguments, and
primitive functions and defined functions (programs) are treated
alike.
The semantic rules are few: the definitions of primitive functions
are independent of the representations of data to which they apply,
all scalar functions are extended to other arrays in the same way
(that is, item-by-item) , and primitive functions have no hidden
effects (so-called side-effects) .
The sequence control is simple: one statement type embraces all
types of branches (conditional, unconditional, computed, etc.), and
the termination of the execution of any function always returns
control to the point of use.
External communication is established by means of variables which
are shared between APL and other systems or subsystems. These
shared variables are treated both syntactically and semantically
like other variables. A subclass of shared variables, system
variables , provides convenient communication between APL programs
and their environment.
The utility of the primitive functions is vastly enhanced by
operators which modify their behavior in a systematic manner. For
example, reduction (denoted by /) modifies a function to apply over
all elements of a list, as in +/L for summation of the items of £.
The remaining operators are scan (running totals, running maxima,
etc.), the axis operator which, for example, allows reduction and
scan to be applied over a specified axis (rows or columns) of a
table, the outer product , which produces tables of values as in
RATE" ,*YEARS for an interest table, and the inner product , a simple
generalization of matrix product which is exceedingly useful in data
processing and other non-mathematical applications.
Section 1 : Introduction 7
The number of primitive functions is small enough that each is
represented by a single easily-read and easily-written symbol, yet
the set of primitives embraces operations from simple addition to
grading (sorting) and formatting. The complete set can be
classified as follows:
Arithmetic : + -xt*«o|L["!S
Boolean and Relational : vava~<<; = 2:>*
Selection and Structural: /\j<\[;] + + p,4>$e
General: e\?iT^iiv
8 APL Language
SECTION 2; FUNDAMENTALS
A typical statement in APL is of the form:
AREA-*-?,***
The effect of the statement is to assign to the name AREA the value of
the expression 3x4 to the right of the specification arrow «- ; the
statement may be read informally as "AREA i£ three times four."
The statement is the normal unit of execution. Two primitive
types occur, the specification illustrated above, and the branch , which
serves to control the sequence in which the statements in a defined
function (discussed in Section 6) are executed. There is also a third
type of statement which may invoke the use of a defined function without
either a specification or a branch.
A variant of the specification statement produces display of a
result on the medium (paper or screen) provided by the terminal device;
if the leftmost part of a statement is not a name followed by a
specification, the result of the expression is displayed. For example:
12
14
3x4
PERIMETER*-2x(3 + H)
PERIMETER
Printing of the result of any part of a statement can be obtained
by including the characters Q+ at the appropriate point in the
statement. Moreover, any number of specification arrows may occur in a
statement. For example:
12
14
4
Xf-2+D-*-3xY-<-4
X
Y
The terminal entry and display devices used with APL systems
include a variety of typewriter-like and display-tube devices. Their
characteristics vary, but the essential common characteristics are:
1. The ability to enter and display APL characters.
2. A means of signalling completion (and release to the system) of an
entry.
3. Facilities for convenient revision of an entry before release.
4. Facilities to interrupt execution at the end of a statement ( weak
interrupt ) and within a statement ( strong interrupt ) .
5. A cursor , such as an arrowhead, to show where on the line the next
character entered will appear.
All examples in this manual are presented as produced on a
typewriter-like device on which the release signal is produced by the
carrier return key, and revision is handled by backspacing to the point
of revision, striking the attention button, and entering the revision.
A caret supplied by the system marks the point of correction. For
Section 2 : Fundamentals
example :
18
3+4x5+
A
+ 5 + 6
On terminals of this type the attention key is also used for
interrupting execution. A single strike of this key while execution is
in progress provides a weak interrupt, and a double strike provides a
strong interrupt.
Entry of a statement which cannot be executed will invoke an error
re P ort . which indicates the nature of the error and the point at which
execution stopped. For example:
Z-f-5
3+(YxX )
VALUE ERROR
3+(7xZ)
A
Error messages are listed in Figure 2.1, together with information on
the cause and suggested corrective action.
CHARACTER SET
The characters which may occur in a statement fall in four main
classes: alphabetic, numeric, special, and blank. The alphabetics
comprise the roman alphabet in upper case italic font, and the same
alphabet underscored. The entire set is shown in Figure 2.2 together
with suggested names and the scheme for forming (as composites of other
symbols) those which may not be directly available on the keys of
certain terminal devices. A typical keyboard layout is shown in Figure
The names suggested are for the symbols themselves and not
necessarily for the functions they represent. For example, the
downstile L represents both the minimum , a function of two arguments,
and the floor (or integer part ) , a function of one argument. In
general, most of the special characters (such as +, -, x, and *) are
used to denote primitive functions which are assigned fixed meanings,
and the alphabetic characters are used to form names which may be
assigned and re-assigned significance as variables, defined functions,
and other objects. The blank serves as a separator to mark divisions
between names (which are of arbitrary length) .
For terminal devices which permit the display font to be changed
without changing the behavior of the entry keyboard or communication
with the system (as in changing the typing element on certain
typewriters), any available display font may be used. For example, in
textual work a font with normal upper- and lower-case roman is commonly
employed.
SCALAR AND VECTOR CONSTANTS
All numbers entered or displayed are in decimal, either in
conventional form (including a decimal point if appropriate) or in
scaled form. The scaled form consists of an integer or decimal fraction
callesd the multiplier followed immediately by the symbol E followed
immediately by an integer (which must not include a decimal point)
called the scale . The scale specifies the power of ten by which the
multiplier is to be multiplied. Thus l . 44E2 is equivalent to 144.
1 APL Language
TYPE
CHARACTER
DOMAIN
DEFN
Q— IMPLICIT
INDEX
INTERFACE
QUOTA
EXHAUSTED
INTERRUPT
LENGTH
NO SHARES
RANK
RESEND
SI DAMAGE
SYNTAX
SYMBOL
TABLE
FULL
SYSTEM
VALUE
WS FULL
Cause; CORRECTIVE ACTION
Illegitimate overstrike.
Arguments not in the domain of the function.
Misuse of V or Q symbols:
1. The function is pendent. DISPLAY STATE INDICATOR AND
CLEAR AS REQUIRED.
2 . Use of other than the function name alone in reopening
a definition.
3. Improper request for a line edit or display.
The system variable D — ( for example, 010) has been set
to an inappropriate value.
Index value out of range.
Attempt to share more variables than allotted quota.
REQUEST LARGER QUOTA FROM APL OPERATOR.
Execution was suspended within an APL statement. TO
RESUME EXECUTION, ENTER A BRANCH TO THE STATEMENT
INTERRUPTED.
Shapes not conformable.
Shared variable facility not in operation.
Ranks not conformable.
Transmission failure or more than the
implementation-allowed number of characters entered in
one line. RE-ENTER. IF CHRONIC, REDIAL OR HAVE TERMINAL
OR PHONE REPAIRED.
The state indicator (an internal list of suspended and
pendent functions) has been damaged in editing a function
or in performing a )COPY or ) ERASE.
Invalid syntax; e.g. two variables juxtaposed; function
used without appropriate arguments as dictated by its
header; unmatched parentheses.
Too many names used. SAVE, CLEAR, COPY
or SAVE, CLEAR, SYMBOLS, COPY
or ERASE, SAVE, CLEAR, COPY
Fault in internal operation of the system. RELOAD. SEND
TYPED RECORD, INCLUDING ALL WORK LEADING TO THE ERROR, TO
THE SYSTEM MANAGER.
Use of name which has not been assigned a value.
A VALUE TO THE VARIABLE, OR DEFINE THE FUNCTION.
ASSIGN
Workspace is filled (perhaps by temporary values produced
in evaluating a compound expression, or by values of
Shared variables) . CLEAR STATE INDICATOR, ERASE NEEDLESS
OBJECTS, OR REVISE CALCULATIONS TO USE LESS SPACE.
Figure 2.1: Error Reports
Section 2 :
Fundamentals
11
A
B C D E F G H
I J
KLMNOPQR
S
T U V W X I Z
A
B C D E F G H
I I
KLMNQPQR
S
nznn
1 2 3 4 5 6 7
8 9
••
dieresis
alpha
V
nor
n+t
V
overbar
r
upstile
H
nand
~
A
<
less
L
downstile
f
del stile
V
|
<
not greater
underbar
4
delta stile
A
I
=
equal
V
del
4>
circle stile
O
I
>
not less
A
delta
*
circle slope
o
\
>
greater
null
e
circle bar
o
*
not equal
t
quote
®
log
o
*
V
or
u
quad
i
I-beam
1
T
A
and
(
open paren
¥
del tilde
V
~
-
bar
)
close paren
4
base null
JL
T
divide
c
open bracket
f
top null
T
o
+
plus
]
close bracket
\
slope bar
\
-
X
times
c
open shoe
/
slash bar
/
-
J>
query
3
close shoe
fi
cap null
n
u
omega
n
cap
13
quote quad
r
D
e
epsilon
u
cup
j
quote dot
t
,
P
rho
i
base
B
domino
n
♦.
~
tilde
T
top
+
up (arrow)
1
stile
+
down (arrow)
9
semicolon
i
iota
:
colon
o
circle
»
comma
*
star
,
dot
■+•
right (arrow)
\
slope
■*-
left (arrow)
/
slash
space
Figure 2.2: The APL Character Set
uu.
<
3
<
4
5
>
6
>
7
8
V
9
A
o
+
•
•
X
BACK
SPACE
-II 5
— — y—
(A)
w
P
R
T
t
Y
I
I
O
o
*
P
.-►
v RETURN
L oc7^[a
r
s
L
D
F
V
A
H
e
J
i
K
D
L
(
E
)
]
SHIFT
c
z
n
c
U
V
±
T
1
M
•
•
*
•
\
/
SHIFT j
Figure 2.3: Typical APL Keyboard
12 APL Language
Negative numbers are represented by an overbar immediately
preceding the number, for example, "1.44 and 144E 2 are equivalent
negative numbers. The overbar can be used only as part of a constant
and is to be distinguished from the bar which denotes negation, as in
-X.
A single numerical constant entered by itself is accepted by the
system as a scalar . A constant vector may be entered by listing the
numerical components in order, separated by one or more spaces. A
scalar character constant may be entered by placing the character
between quotation marks, and a character vector may be entered by
listing the characters between quotation marks. Such a vector is
displayed by the system as the sequence of characters, with no enclosing
quotes and with no separation of the successive elements. The quote
character itself must be entered as a pair of quotes. Thus, the
abbreviation of CANNOT is entered as x CAN xx T r and prints as CAN'T.
SPACES
The blank character is used primarily as a separator. The spaces
that one or more blank characters produce are needed to separate names
of adjacent defined functions, constants, and variables. For example,
if F is a defined function, then the expression 3 F 4 must be entered
with the indicated spaces. The exact number of spaces used in
succession is of no importance, and extra spaces may be used freely.
Spaces are not required between primitive functions and constants or
variables, or between a succession of primitive functions, but they may
be used if desired. For example, the expression 3+4 may be entered with
no spaces.
FUNCTIONS
The word "function" derives from a word which means to execute or
to perform. A function executes some action on its argument (or
arguments) to produce a result which may serve as an argument to another
function. For example: ~ ~"
12
14
"2
3x4
2+(3x4)
(-6)t3
A function (such as the negation used above) which takes one
argument is said to be monadic , and a function (such as times ) which
takes two arguments is said to be dyadic . All APL functions are either
monadic or dyadic or, in the case of defined functions only, niladic
(taking no argument) . Certain of the special symbols are used to denote
two different functions, one monadic and the other dyadic. For example,
X-Y denotes s ubtraction of Y from X (a dyadic function) , and -Y denotes
negation of I (a monadic function) . Other examples appear in Figures
3 . 1 and 3.5.
Each of the primitive functions is denoted by a single character
or by an operator (discussed in Section 3) applied to such a character.
For example, + and * are primitive functions as are +/ and * / (since /
denotes an operator) .
Section 2 : Fundamentals 1 3
ORDER OF EXECUTION
Parentheses are used in the familiar way to control the order of
execution in a statement. Any expression within matching parentheses is
evaluated before applying to the result any function outside the
matching pair.
In conventional notation, the order of execution of an
unparenthesized sequence of monadic functions may be stated as follows:
the (right-hand) argument of any function is the value of the entire
expression to the right. For example, Log Sin Arctan x means the Log of
Sin Arctan x, which means Log of Sin of Arctan x. In APL, the same rule
applies to dyadic functions as well. Moreover, all functions, both
primitive and defined, are treated alike; there is no hierarchy among
functions (such as multiplication being done before addition or
subtraction) .
An equivalent statement of this rule is that an unparenthesized
expression is evaluated in order from right to left. For example, the
expression 3x8T3*|5-7 is equivalent to 3x(8["(3*(l(5-7)))). Their result
is 27. A consequence of the rule is that the only substantive use of
parentheses is to form the left argument of a function. For example,
(12t3)x2 is 8 and 12*3*2 is 2. However, redundant pairs of parentheses
can be used at will. Thus, 12* (3*2) is also 2.
ARRAYS
APL functions apply to collections of individual items called
arrays . Arrays range from scalars, which are dimensionless, to
multi-dimensional arrays of arbitrary rank and size.
The vector is a simple form of array which may be formed by
listing its elements as described in the discussion of constants. For
example:
V+2 35711131719
A+ X ABCDEFGR X
The elements of a vector may be selected by indexing . For example:
F[3 1 5]
5 2 11
AlQ 5 14]
HEAD
Arrays of more complex structure may be formed with the reshape
function denoted by p:
M<-2 4p7 S+-2 UpA
M B
2 3 5 7 ABCD
11 13 17 19 EFGH
These results have two dimensions or axes and are called t able s or
matrices . A matrix has two axes "and is said to be of rank 2 ; a vector
has one axis and is of rank 1 .
The left argument 2 4 in the foregoing examples specifies the
shape of the resulting array. Arrays of arbitrary shape and rank may be
1 >i APL Language
produced by the same scheme. For example:
T+2 3 Hp'ABCDEFGHIJKLMNOPQRSTUVWX x
T
ABCD
EFGE
IJKL
MNOP
QRST
UVWX
The shape of an array can be determined by the monadic function
denoted by p :
p7 pA? pT
8 2 4 2 3 4
Elements may be selected from any array (other than a scalar) by
indexing in the manner shown for vectors, except that indices must be
provided for each axis:
M[2;3] T[2;l;4]
17 P
Ml 2 1;2 3 4] Tl2;l 2 3;1 2 3 4]
13 17 19 MNOP
3 5 7 QRST
U7WX
The indexing used in the foregoing examples is called 1 - origin
because the first element along each axis is selected by the index i .
One may also use o-origin indexing by setting the index v origin to o.
The index origin is a system variable (See Section 4) denoted by 010.
Thus:
QI0+1 UIO+-0
711 2 3] y[0 1 2]
2 3 5 2 3 5
3[2;3] B[l;2]
G G
All further examples assume i-origin unless otherwise stated.
WORKSPACES AND LIBRARIES
The common organizational unit in an APL system is the workspace .
When in use, a workspace is said to be active , and it occupies a block
of working storage in the central computer. Part of each workspace is
set aside to serve the internal workings of the system, and the
remainder is used, as required, for storing items of information and for
containing transient information generated in the course of a
computation.
An active workspace is always associated with a terminal during a
work session, and all transactions with the system are mediated by it.
In particular, the names of variables (data items) and defined functions
(programs) used in calculations always refer to objects known by those
names in the active workspace; information on the progress of program
execution is maintained in the state indicator of the active workspace;
and control information affecting the form of output is held within the
active workspace.
Section 2 : Fundamentals 1 5
Inactive workspaces are stored in libraries , where they are
identified by arbitrary names. They occupy space in secondary storage
facilities of the central computer and cannot be worked with directly.
When required, copies of stored workspaces can be made active, or
selected information may be copied from them into an active workspace.
Libraries are either private or public . Private libraries are
associated with individual users of the system, and are identified by
the user's account number. Access to them by other users is restricted
in that one user may not store workspaces in another person's library,
nor obtain a listing of the workspaces already stored there. However,
one user may activate a copy of another user's unlocked workspace if he
knows the library number and workspace name. Workspaces and libraries
are managed by system commands as described in Section 8 .
NAMES
Names of workspaces, functions, variables, and groups (see Section
8) may be formed of any sequence of alphabetic and numeric characters
that starts with an alphabetic and contains no blank.
The environment in which APL operations take place is bounded by
the active workspace. Hence, the same name may be used to designate
different objects (that is, groups, functions, or variables) in
different workspaces, without interference. Also, since workspaces
themselves are never the subject of APL operations, but only of system
commands, it is possible for a workspace to have the same name as an
object it holds.
Stored workspaces and the information they hold can be protected
against unauthorized use by associating a lock , comprising a colon and a
password of the user's choice, with the name of the workspace, when the
workspace is stored. In order to activate a locked workspace or copy
any information it contains, a colon and the password must again be
used, as a key , in conjunction with the workspace name. Listings of
workspace names, including those in public libraries, never give the
keys, and do not overtly indicate the existence of a lock.
USER IDENTIFICATION
Account numbers can be similarly protected by locks and keys, thus
maintaining the security of a user's private library and avoiding
unauthorized charges against his account. Passwords for locks and keys
may be formed of any sequence of alphabetic and numeric characters . In
use as either a lock or key, a password follows the number or name it is
protecting, from which it is set off by a colon.
APL systems differ in the procedure required to sign-on or begin a
session. However, a numerical identification of the user is usually
provided in some manner, so that it can be used to specify the first
element of DAI.
16 APL Language
SECTION 3: PRIMITIVE FUNCTIONS AND OPERATORS
The primitive functions fall in two classes, scalar , and mixed .
Scalar functions are defined on scalar arguments and are extended to
other arrays item-by-item. Mixed functions are defined on arrays of
various ranks and may yield results which differ from the arguments in
both rank and shape. Five primitive operators apply to scalar dyadic
functions and to certain mixed functions to produce a large number of
new functions.
The definitions of certain functions depend on certain system
variables whose names begin with the symbol Q (as in QIO and UCT) .
These system variables are discussed further in Section 4.
SCALAR FUNCTIONS
A monadic scalar function extends to each item of an array
argument; the result is an array of the same shape as the argument and
each item of the result is obtained as the monadic function applied to
the corresponding item of the argument.
A dyadic scalar function extends similarly to a pair of arguments
of the same shape. To be conformable, the arguments must agree in
shape, or at least one of them must be a scalar or one-element vector.
If one of the arguments has only one item, that item is applied in
determining each element of the result. For example:
1 2 3x4 5 6
4 10 18
3 + U 5 6
7 8 9
2 3+4 5 6
LENGTH ERROR
Each of the scalar functions is defined on all real numbers with
two general exceptions: the four boolean functions are defined only on
the numbers o and l, and the functions = and * are defined on characters
as well as numbers. Specific exceptions (such as 4-fO) will be noted
where appropriate .
The scalar functions are summarized in Figure 3.1 together with
their symbols and brief definitions or examples which should make their
use clear. The remainder of this section is devoted to more detailed
definitions.
A dyadic function f may possess a left identity element L such
that L f X equals X for any J, or a right identity element R such that
X f R equals X. For example, one is a right identity element of * since
XtI is X, zero is a left or right identity of +, one is a left or right
identity of x , and the general logarithm function ® has no identity
element.
Identity elements become important as the appropriate result of
applying a function over an empty vector; for example, the sum over an
empty vector is 0, (the identity element of +) , and the product over an
empty vector is 1, the identity element of x. These matters are
discussed further in the treatment of the reduction operator which
concerns such applications of dyadic functions over vectors.
Figure 3.2 lists the identity elements of the dyadic scalar
functions. The relational functions <, £, =, £, > and * possess no true
identity elements, but do when considered as boolean functions, that is,
i
Section 3 : Primitive Functions and Operators 1 7
Monadic form ffl
Dyadic form AfB
Definition
or example
Name
Name
Definition
or example
+B IS B
-B is 0-B
xB is (B>0)-B<0
*B is ltB
|~3.14 is 3 .14
B
LB
rs
3. 14
"3.14
3
~4
4
"3
?B is Random choice
from iB
*B is (2 .71828. . )*B
»*N is W is *«N
OB is Bx3 . 14159 . . .
!0 is 1
!B is Bx!B-l
or IB is Gamma (B+l)
"1 is
'0 is l
Conjugate
Negative
Signum
Reciprocal
Magnitude
Floor
Ceiling
Roll
Exponential
Natural
logarithm
Pi times
Factorial
Not
{-A)QB
A
4 OB
(1-B*2)* . 5
(1-3*2)*. 5
Arcs in B
1
Sine B
Arccos B
2
Cosine B
Arctan B
3
Tangent B
("l+B*2)*.5
4
(1+B*2)*.5
Arsinh B
5
Sinh B
Arcosh B
6
Cosh B
Artanh B
7
Tanh B
Table of Dyadic o Functions
Plus
Minus
Times
Divide
Residue
Minimum
Maximum
Deal
Power
General
logarithm
2+3. 2 IS 5.2
2-3 .2 is "1.2
2x3.2 is 6.4
2*3 . 2 is .625
A \B is B-A-*LB*A+A =
3L7 is 3
3T7 is 7
A Mixed Function (See
Figure 3.5)
2*3 is 8
A»B is Log B base A
A®B is (&B)t®A
Circular, Hyperbolic, Pythagorean
(See table at left)
Binomial
And
Or
Nand
Nor
Less
Not greater
Equal
Not less
Greater
Not Equal
A\B is (!B)*(!i4)x!B-i4
2!5 is 10 3!5 is 10
AVB
A
B
AAB
AVB
A«B
1
1
1
1
1
1
1
1
1
1
1
Relations
Result is 1 if the
relation holds ,
if it does not:
3<7 is 1
7<3 is
Figure 3.1: Primitive Scalar Functions
1 8 APL Language
when restricted to the domain
included in the figure.
and i. These identity elements are
Dyadic
Identity
Left-
Function
Element
Right
Plus
<
L R
Minus
-
R
Times
X
1
L R
Divide
■f
1
R
Residue
1
L
Minimum
L
(note 1 )
L R
Maximum
r
(note 2)
L R
Power
*
1
R
Logarithm
®
None
Circle
o
None
Binomial
1
1
L
And
A
1
L R
Or
V
L R
Nand
i\
None
Nor
¥
None
Less
<
\ Apply
L
Not greater
<
1 I for
L
Equal
=
1 \ boolean
L R
Not less
>
l ( arguments
R
Greater
>
\ only
R
Not equal
£
o)
L R
Note 1 : the largest representable
number
Note 2 : the greatest in magnitude
of representable negative numbers
Figure 3.2:
Identity Elements of
Primitive Scalar
Dyadic Functions
PLUS, MINUS, TIMES, DIVIDE AND RESIDUE
The definitions of the first four of these functions agree with
the familiar definitions except that the indeterminate case 0*0 is
defined to yield the value 1. For X*o , the expression ;sr*0 evokes a
domain error.
If A and B are positive integers, then the residue A\B is the
remainder on dividing A into B. The following definition covers all
values of A and B:
If A=0 , then A\B equals B
If A*Q, then A\B lies between A and zero (being permitted to equal
zero but not A) and is equal to B-N*A for some integer N.
For example:
1| 2.385
I 5,
0.385
5.8
12
3 I 3 2 10123
"1 ~2 "1
3|~3 "2 "1 1 2 3
12
Section 3 : Primative Functions and Operators 1 9
CONJUGATE, NEGATIVE, SIGNUM, RECIPROCAL, MAGNITUDE
The conjugate function +x yields its argument unchanged, the
negative function -X yields the argument reversed in sign, and the
reciprocal *X is equivalent to nx. For example, if x«-4 ""5, then:
+X -X iX
4-5 "4 5 0.25 "0.2
The result of the sign urn function *x depends on the sign of its
argument, being "1 if x<0 , and if 1=0, and l if x>0. The magnitude
function \X (also called absolute value ) yields the greater of X and -X;
in terms of the signum function it is equivalent to XxxX. For example:
x~3 4 |"3 4
10 1 3 4
BOOLEAN AND RELATIONAL FUNCTIONS
The boolean functions and , or , nand (not-and) , and nor (not-or)
apply only to boolean arguments, that is, and 1; if is interpreted
as false , and 1 as true , then the definitions of these functions are
evident from their "names. For example, ,4 AS (read as A and B) equals 1
(is true ) only if A equals 1 (is true ) and B equals 1. All cases are
covered by the following examples:
A+0 Oil
fl+0 10 1
A*B avB A*B A*B
0001 0111 1110 1000
The monadic function not yields the logical complement of its
argument, that is ~0 is 1, and ~l is 0.
The relational functions apply to any numbers, but yield only
boolean results, that is or 1. The result is l if the indicated
relation holds, and otherwise. For example:
3 5<5 3 3 5 7*7 5 3
10 10 1
The comparisons involved in determining the results of the
relational functions are not absolute, but are made to a certain
tolerance specified by the comparison tolerance OCT. Two scalar
quantities A and B are considered to be equal if the magnitude of their
difference does not exceed the value of UCT multiplied by the larger of
the magnitudes of A and B, that is, if (\A-B) is less than or equal to
Uct*{ u>r |s.
Similarly, A>B is considered to be true if (.A-B) is greater than
or equal to -QCTx( U)f( |B) , and A>B is considered true if AZB is true
and A=B is not.
The comparison tolerance OCT is typically set to the value 1S~13.
The setting OCT+Q is also useful since it yields absolute comparisons,
but may lead to unexpected results due to the finite precision of the
representation of numbers. For example, if the maximum precision is 16
20 APL Language
decimal digits and all digits are displayed in printing, then:
QCT+O
X
0.6666666666666667
Y-f-3x*
1-2
2.2204460'+9250313£' 16
2 = 1
0CT+1E 13
2 = Y
1
When applied to boolean arguments only, the relations are, in
effect, boolean functions, and denote functions which may be familiar
from the study of logic, although referred to by different names and
symbols. For example, X*Y is the exclusiye- or of X and Y, and XzY is
material implication . This association should" be clear from the
following table, which lists in the first two columns the four possible
sets of values of two boolean arguments, and in the remaining columns
the values of the 16 possible boolean functions, with the symbols of the
boolean and relational functions of APL appended to appropriate columns :
A B A f B
0000000011111111
01 0000111100001111
10 0011001100110011
11 01010101010 10101
A > < * V V = £ <. *<
The ten functions listed at the foot of this table embrace all
non-trivial boolean functions of two arguments. Consequently, any
boolean expression of two arguments X and Y can be replaced by a simple
APL expression as follows: evaluate the expression for the four
possible cases, locate the corresponding column in the table, then use
the function symbol at the foot of the column, or, if none occurs, use X
or Y or ~X or ~Y or or 1 as appropriate.
MINIMUM AND MAXIMUM
The functions denoted by L and r perform as expected from their
names. For example:
X+~3 "2 "1 0. 1 2 3
Y+-3 2 1 "1 "2 "3
X[Y
3 2 10 12 3
XlY
"3 2 "1 "1 "2 ~3
FLOOR AND CEILING
The floor function L yields the integer part of its argument, that
is, IX yields the largest integer which does not exceed X. Similarly,
[X yields the smallest integer which is not less than X. For example:
X-f-~3.lt 2.718
IX \X
~4 2 "3 3
"4 2 ~3 3
.Section 3: Primative Functions and Operators 21
The ceiling and floor functions are affected by the comparison
tolerance OCT as follows: if there is an integer I for which \X-I does
not exceed the value of OCT, then both IX and {X equal I. For example,
if results are represented and printed to 16 decimal digits, then:
J«-3x2*3
QCT+1E~13
ncT+o
IX
2
IX
[X
3
\X
ROLL (Random number function)
The roll is a monadic function named by analogy with the roll of a
die; thus ?6 yields a (pseudo-) random choice from i"6, that is the first
six integers beginning- with either or l according to the value of the
index, origin UIO. For example:
UI0+-1
? 6 ? 6 ? 6
15 3
?6 666666666666
4215563 4 5 114 5
UIO+O
? 6666666666666
0202435 5 30324
The domain of the roll function is limited to positive integers.
The roll function employs an algorithm due to D.H. Lehmer; the
result for each scalar argument X is a function of X and of the random
link variable URL. The result of the roll function is system dependent
but typically for X<2*31 is equal to UIO plus the integer part of
X*URL*~1+2*31.
POWER, EXPONENTIAL, GENERAL AND NATURAL LOGARITHM
For non-negative integer right arguments the power function X*N is
simply defined as the product over N repetitions of X. It is
generalized to non-positive and non-integer arguments so as to preserve
the relation that X*A+B shall equal (X*A )x(X*B ) . Familiar consequences
of this extension are that x*-N is the reciprocal of X*N, and x**N is
the Nth root of X. For example:
2*~3 "2 "1 1 2 3
.125 .25 .51248
64**1 2 3 4 5 6
64 8 4 2.828 2.297 2
The indeterminate case 0*0 is defined to have the value l.
The domain of the power function X*Y is restricted in two ways:
if Z=0, then Y must be non-negative; if X<0, then Y must be an integer
or a (close approximation to a) rational number with an odd denominator.
For example, ~8*.5 yields a domain error, but ~8*1*3 and ~8*2*3 yield ~2
and 4, respectively.
The exponential function *X is equivalent to the expression e*X,
where e is the base of the natural logarithms, approximately 2, .71828.
22 APL Language
For example:
*~2 "10 12
0.1353352832 . 367 879441 2 1 2.718281828 7.389056099
The natural logarithm function ®X is the inverse of the
exponential, that is, *®X and ®*X both equal X. For example:
• 1 2 3 4
0.6931471806 1.098612289 1.386294361
r *®l 2 3 4
12 3 4
«*1 2 3 4
12 3 4
The domain of the natural logarithm function is limited to
positive numbers.
The general logarithm function B*X is defined as (®X)*®B. It is
inverse to the power function in the following sense: B*B*X and B®B*X
both equal X. Limitations on the domain follow directly from the
defining expression.
CIRCULAR, HYPERBOLIC, AND PYTHAGOREAN FUNCTIONS
The symbol o denotes a monadic function whose result equals pi
times its argument. For example:
oi 2 .5
3.141592654 6.283185307 1.570796327
The symbol o is also used dyadically to denote a family of fifteen
related functions as follows: the expression iox is defined for integer
values of I from "7 to 7, and is in each case equivalent to one of the
circular, hyperbolic, or pythagorean functions as indicated in Figure
3.1.
The arguments for the circular functions Sin, Cos, and Tan (iox,
2oX, and 3oz) are in radians. For example:
PJ-HDl
10PI*2 3 4
1 0.8660254038 0.7071067812
The hyperbolic functions sinh and cosh are the odd and even
components of the exponential function ; . that is, 5oX is odd, 60* is
even, and the sum (5oZ)+(6oZ) is equivalent to *X. Consequently:
5oX equals . 5x(*X) -i*-X)
&oX equals . 5x( *X)+(*-X)
The definition of the hyperbolic tangent is analogous to that of the
tangent, that is, 7ox equals (5oX)*6oX.
The. three pythagorean functions oox, 4ox, and ~4oX are defined as
in Figure 3.1, and are related to the properties of a right triangle as
Section 3: Primative Functions and Operators 23
indicated in Figure 3.3. They may also be defined as follows:
"40X equals
oox equals
40X equals
50
20
6O
6 OX
'lox or
'50X
10 20X
1 A >^
E
D
AC-\
AB=OOBC
BC=OOAB
AE=^ODE
DE- koAE
I B
The Pythagorean Functions
Figure 3.3
Each of the family of functions lox has an inverse in the family,
that is, (-I)oX is the inverse of lox. Certain of the functions are not
monotonic, and their inverses are therefore many-valued. The principal
values are chosen in the following intervals:
Arcosh
i?«-~60X
i?S0
R+ 4oX
J? 2
Arccos
R+ 2 ox
(i?S0)A(i?<Ol)
Arc sin
R*- lox
( |fl)<o.5
R+- OOX
i?20
f?«- 4 OX
i?>0
FACTORIAL AND BINOMIAL FUNCTIONS
For positive integer arguments, the factorial function \N is
defined as the product of all positive integers up to N. An important
consequence of this definition is that IN equals ffx'.jy-i, or,
equivalently IN-l equals (IN)*N. This relation is used to extend the
function to non-integer and negative arguments. For example:
ff+1 2 3 4 5
IN
1 2 6 24- 120
( IN)*N
1 1 2 6 24
!0 1 2 3 4
1 1 2 6 24
Ft- . 5 1 1.5 2 2.5
IF
0.8862269255 1 1.329340388 2 3.32335097
( \F)iF
1.772453851 1 0.8862269255 1 1.329340388
!~.5 .5 1 1.5
1.772453851 1 0.8862269255 1 1.329340388
This extension leads to the expression (ro)*0, or 1*0 for ! "1 , and
"1 is therefore excluded from the domain of factorial, as are all
negative integers.
24 APL Language
The binomial function M\N is defined, for non-negative integer
arguments, as the number of distinct ways in which M things can be
chosen from N things. The expression ( \N)i ( !M)x( IN-M) yields an
equivalent definition which is used to extend the definition to all
numbers. Although the domain of factorial excludes negative integers,
the domain of the binomial does not, since any implied division by zero
in the numerator \N is invariably accompanied by a corresponding
division by zero in the denominator; the function therefore extends
smoothly to all numbers.
The result of UN is equivalent to coefficient I in the binomial
expansion (X+1)*N . For example:
1 2 3 !3
13 3 1
OPERATORS
An operator may be applied to a function to yield a different
function. For example, the outer product operator denoted by the
symbols ° . may be applied to any of the primitive scalar dyadic
functions to produce a corresponding "table function" as illustrated
below for times and power ;
A+l 2 3
A° . xA
2 3 4
i+6 8
6 9 12
8 12 16
A» .*A
111
4 8 16
9 27 81
16 64 256
Four of the APL operators - reduction , scan , inner product , and
outer product - apply to any primitive scalar dyadic function. The axis
operator applies to reduction and scan , and also to certain of the
mixed functions.
REDUCTION
Reduction is denoted by the symbol / and applies to the function
which precedes it. For example, if V+-1 2 3 4 5, then +/V yields the sum
of the items of V, and x/7 yields their product:
t/V
x/P
15
120
In general, an expression of the form f/V is equivalent to the
expression obtained by placing the function symbol f between adjacent
pairs of items of the vector V:
r/v
-/V
ir2r3T4r5
1-2-3-4-5
The last example points up the fact that the general rule for the
order of execution is applied, and that as a consequence the expression
-/V yields the alternating sum of the items of V. The alternating sum
is defined as the sum obtained after first weighting the items by
Section 3: Primative Functions and Operators 25
multiplying alternate elements by l and l . Thus :
A+l ~1 1 "1 1
VxA
12 3 4 5
+ /VxA
■IV
3
3
Similarly, r/V yields the alternating product ;
V*A
1 .53 .25 5
x/V*A
1.875
*/V
1.875
The result of applying reduction to any scalar or vector is a
scalar; the value in the case of a scalar or one-element vector argument
is the single item itself. The application of reduction to other arrays
is treated in the discussion of the axis operator.
Reduction of an empty vector by any function is defined as the
identity element of the function if one exists, i&§> a domain error if
not. Thus if V is an empty vector, +/V equals 0, and a/7 equals 1.
The reason for this definition is the extension to empty vectors
of an important relation between the reductions of two vectors P and Q
and the reduction of the vector V+P.Q obtained by chaining them
together. For example:
+/7 equals (+/P) + (+/<2)
x-/V equals (x/p)x(x/Q)
If P is an empty vector it is clear that +/P must equal (the identity
element of +) , and that x/p must equal 1.
SCAN
The scan operator is denoted by the symbol \ and applies to the
function which precedes it. When the resulting function is applied to a
vector V it yields a vector of the same shape whose Kth element is equal
to the corresponding reduction over the first K elements of V. For
example :
+\1 2 3 4 5
1 3 6 10 15
x\l 2 3 4 5
1 2 6 24 120
V\0 10 1
111
A\l 10 10
110
<\0 10 110
10
The extension of scan to arrays other than vectors is treated in
the discussion of the axis operator.
26 APL Language
AXIS
Since reduction is defined on vectors, and since a matrix can be
conceived as a collection of row vectors, the result of reduction on a
matrix could be defined as the vector of results obtained by reduction
of each of the row vectors. However, a matrix can also be viewed as a
collection of column vectors, and reduction could be defined
accordingly. It is therefore important to be able to specify which set
of vectors is to be reduced.
The row vectors extend along the second axis of the matrix, and
are each of length equal to the length of that axis. Similarly, the
column vectors extend along the first axis and, in general, the index of
an axis identifies a set of vectors in an array of any non-zero rank.
The axis operator is denoted by brackets enclosing an expression
which yields the index of an axis as a one-element vector or a scalar.
It determines the direction of application of any scan or reduction
operator whose symbols immediately precede it. For example:
M-e-3 4pil2 +\[1]M +\[2]M
1234 1234 136 10
5/678 6 8 10 12 5 11 18 26
9 10 11 12 15 18 21 24 9 19 30 42
+/111M
15 18 21 24
+/L21M
10 26 42
The shape of the result of scan equals the shape of the argument.
The shape of the result of reduction equals the shape of the argument
except that the indicated axis of reduction is removed. The indexing of
axes depends on the index origin 010.
If no axis operator occurs, reduction and scan apply along the
last axis. The symbols / and \ also denote reduction and scan, but in
the absence of axis operators they apply along the first axis .
The axis operator is also used to specify the axis of application
of the mixed functions reverse , rotate , compress , and expand .
INNER PRODUCT
If P and Q are vectors of the same' shape, then the expression
+/PxQ has a variety of useful interpretations. For example, if P is a
list of prices and Q a list of corresponding order quantities, then
+/PxQ is the total cost. Expressions of the same form employing
functions other than + and x are equally useful, as suggested by the
following examples (where B is used to denote a boolean vector} :
a/P=Q Comparison of P and Q.
+/P=Q Count of agreements between P and Q.
l/P+Q Minimum distance for shipment, where P represents the
distances from source to transhipment points and Q the
distances from these points to the destination.
+/PxB Sum over a subset of P specified by B.
*/P*B Product over a subset of P specified by B.
The inner product produces functions equivalent to expressions in
this form; it is denoted by. a dot and applies to the two functions which
surround it. Thus P+.xQ is equivalent to +/Px#, and Px.*B is equivalent
Section 3: Primative Functions and Operators 27
to x/p*p, and, in general, Pf.gC is equivalent to f/PgQ, if P and Q are
vectors.
The inner product is extended to arrays other than vectors along
certain fixed axes, namely the last axis of the first argument and the
first axis of the last argument. The lengths of these axes must agree.
The sh ape of the result is obtained by deleting these axes and chaining
the remaining shape vectors together. The consequence for matrix
arguments is illustrated in Figure 3.4. The consequences for the shape
of inner products on some other arrays are illustrated below:
Q.G Q.H
7 7
( scalar )
pGf.gH
Q.A B.S.
3 5 ^5 2 7
7 9 9
1 ^
3 2 7
pAf.gB
7
pCf .go
pEf.gF
Formally, pAf.gB equals ( 1+pA) ,1+pB.
The inner product M+.*N is commonly called the matrix product.
Examples of it and other inner products follow:
P«-2 3 5 7
M W+ . xM
1111 1234
0111 0123
0011 0012
0001 0001
M+.*P
17 15 12 7
P+ . x-M
2 5 10 17
MA.=Af
1
Px . *M
2 6 30 210
M- . xM
1 _0 1 _0
10 1
1 _o
1
MA.
10
= 0011
Figure 3.4: Inner Product
28 APL Language
Either argument of an inner product may be a scalar; it is
extended in the usual manner. For example, A+.*l is equivalent to
+ /A, and l+.xA is equivalent to +/A.
OUTER PRODUCT
The outer product operator, denoted by the symbols °. preceding
the function symbol, applies to any dyadic primitive scalar function,
so that the function is evaluated for each member of the left argument
paired with each member of the right argument. For example, if
ii-f-1 2 3 and S-e-l 2 3 4 5:
A" . xB
12 3 4 5
2 4 6 8 10
3 6 9 12 15
Ao ,<B
1111
111
11
Such tables may be better understood if labelled in a way widely used in
elementary arithmetic texts: values of the arguments are placed beside
and above the table, and the function whose outer product is being
computed is shown at the corner. Thus:
B
B
* 1
1
1
2
2
3
3
4
4
5
5
< 1
1
2
3
4
5
1 1
1 1
1
1
1
1
2 1
2
4
6
8
10
A
2 1
1
1
1
3 1
3
6
9
12
15
3 1
1
1
In the foregoing example the shape of the result A°.*B is clearly
equal to (piO.(pS). This expression yields the shape for any arguments
A and B. Thus if R+-A°.+B and A is a matrix of shape 3 4 and B is a
three-dimensional array of shape 5 6 7, then R is a five-dimensional
array of shape 3 4 5 6 7. Moreover, RH ;J ;K;L;M1 equals ALI;J1+BLK;L ;Ml
for all possible values of the indices.
MIXED FUNCTIONS
The mixed functions are grouped in five classes according to
whether they concern the structure of arrays , selection from arrays , the
generation of selector information for use by selection functions,
numerical calculations, or transformations of data such as that between
characters and numbers. All are listed in Figure 3.5 together with
brief definitions or examples.
Those functions which may be modified by an axis operator may also
be used without an axis operator, in which case the axis concerned is
the last, or, in the case of the functions denoted by e and by /, the
first.
Section 3: Primative Functions and Operators 29
Name
Sign
Definition or example
Shape
Reshape
Ravel
Reverse
Rotate s
Catenate
Transpose 3
pA
VqA
,A
$A
Ai>A
A, A
VQA
QA
p p is n p e is 3 4 p5 is i0
Reshape a to dimension v 3 4pil2 is E
12pE is 112 OpE is i0
,A is (y./pA)pA ,E is il2 p,5 is 1
DCS A IJKL
W is HGFE *[1]X is ex is EFGH
LKJI $P is 7 5 3 2 /1BCD
3<t>P is 7 2 3 5 is ~l<t>f> 1 ~1(|>X is SFGH
P,i2 is 235712 <T','HIS' is 'THIS'
AEI
Coordinate I of 4 2 IS)* is BFJ
becomes coordinate CGK
VLI] of result i i$£ is l 6 ll DHL
Reverse order of coordinates §E is 2 lS)ff
Take
Drop
Compress
Expand
Indexing
V+A
V\A
VIA
V\A
VIA1
MIA; A]
ALA;. .
..;4]
} Take or drop |f[I] first 2 3+X is 4SC
> (Vlllzo) or last (F[I]<0) EFG
) elements of coordinate j ~2+P is 5 7
1 3
10 1 0/P is 2 5 10 1 0/E is 5 7
9 11
1 1/[1]£ is 1 2 3 1+ is 1 HE
9 10 11 12
A BCD
1 l\i2 is 102 1011 1\X is E FGH
I JKL
P[2] is 3 P'[ 4 3 2 1] is 7 5 3 2
E[l 3;3 2 1] is 3 2 1
11 10 9
E[l;] is 1 2 3 4 .4BCD
E[;l] is 1 5 9 ' ABCDEFGHIJKL' [E] is EPfffl
Index:
generator
Index of
Membership
Grade up *
Grade down
Deal"
\S
V\A
AeA
kv
yv
s?s
First s integers it is l 2 3 4
i0 is an empty vector
Least index of A Pi 3 is 2 5 12 5
in v, or l+pf PxE is 3 5 4 5
4 4l4 is 1 5 5 5 5
110
pWeY is B W EsP is l o 1
Peil is 1 1 0000
■) The permutation which A3532is4l32
Would order v (ascend-
) ing or descending) f3 532 is 2134
W?X is Random deal of w elements from iY
Matrix
inverse
Matrix
division
Decode
Encode
SM
MMM
AxA
ATA
B2 2pl 1 1 is l ~i arguments may
1 be scalars,
(2 2pP)@2 2pl l o l is ~3 "4 vectors, or
5 7 matrices
1011 7 7 6 is 1776 24 60 60ll 2 3 is 3723
24 60 60T3723 is 1 2 3 60 60T3723 LB 2 3
Execute
Format ,
monadic
Format ,
dyadic
i.V
VIA
l'lt2' is 3 i'P' is 2 3 5 7
'~1 .5'a.=t~1 . 5 is 1
pTE is 3 12 X is jX
UP is 4 1TP is 2.0 3.0 5.0 7.0
~1TP is 6 "lfP is 2E00 3E00 5E00 7EO0
Notes to Figure 3.5
1 . Restrictions on
argument ranks are
indicated by: S for
scalar, V for vector,
M for matrix, and A
for any array.
Except as the left
argument of Si A or
SZA1 , or the argument
of -i and t, a scalar
may be used in place
of a vector. A
one-element vector
may replace any
scalar.
2. Arrays used in
examples:
P
2 3 5 7
E
12 3 4
5 6 7 8
9 10 11 12
X
ABCD
EFGH
IJKL
3. Function depends
on index origin.
t. Elision of any
index selects all
along that axis.
5. The function is
applied along the
last axis; the
symbols 1, \, and e
are equivalent to /,
\, and if,
respectively, except
that the function is
applied along the
first axis. In
general, the relevant
axis is determined by
[VI after the
function symbol.
Figure 3.5: Primitive Mixed Functions
30 APL Language
A scalar may be used in place of a one-element vector:
a. as left argument of
reshape 3 pit -*--* (,3)p^
take 3 -M 5 ■*--»■ ( , 3 ) 1 1 5
drop 34-i5 «-»■ ( , 3 ) + 1 5
expand 1 \ . 5 ■<--»■ ( , 1 ) \ , 5
transpose IS?, 5 ■*■■*■ (,1)<?,5
format 5f3.2 +•+ (,5)T3.2 «-+ 5 T3.2
b. as right argument of
execute *'P' ■*■+ 4,'P 1
branch -»-4 ■<-•*• '-*-,4
A scalar is extended to conform to a vector:
a. as left argument of
compress 1/ i3 ++ 1 1 1 / i3
rotate l<t>2 2 pit +■-* 1 1 <J> 2 2pi<+
b. as right argument of
compress 101/2 •*-->■ 101/222
expand 1 1 \ 2 «--*• 1 1 \ 2 2
A one-element vector is permitted in place of a scalar:
a. as left argument of
deal (,3)?5 ++ 3?5
b. as right argument of
index generator i , 5 ■*--+ i 5
deal 3?, 5 •*--* 3?5
Figure 3.6: Scalar-Vector Substitutions for Mixed Functions
Figure 3.5 gives restrictions, in summary form, on the ranks of
arguments which may be used with each mixed function. Figure 3.6 shows
for what mixed functions and under what conditions scalar and vector
arguments may be substituted for each other.
STRUCTURAL FUNCTIONS
In the monadic structure functions the argument may be of any
type, numeric or character. In the dyadic selection and structure
functions one argument may be of any type and the other (which serves as
an index or other selection indicator) must be numeric, and xn two cases
(compression and expansion) is further restricted to be boolean.
Shape Reshape and Ravel
The monadic function p applied to an array A yields the shape of
A, that is, a vector whose components are the dimensions of A. For
Section 3: Primative Functions and Operators 31
example, if A is the matrix
2 3
6 7
10 11
12
of three rows and four columns, then pA is the vector 3 4.
Since pA contains one component for each axis of A, the expression
ppA is the rank of A. Figure 3.7 illustrates the values of pA and ppA
for arrays of rank o (scalars) up to rank 3. In particular, the
function p applied to a scalar yields an empty vector .
Type of Array
pA
ppA
pppA
Scalar
Vector
Matrix
3-Dimensional
N
M N
L M N
1
2
3
1
1
1
1
Figure 3.7 SHAPE AND RANK VECTORS
The monadic function ravel is denoted by a comma; when applied to
any array A it produces a vector whose elements are the elements of A in
row order. For example, if A is the matrix:
2
10
18
4
12
20
6
14
22
16
24
and if V+,A , then V is a vector of dimension 12 whose elements are the
integers 2 4 6 8 10 12 ... 24- If A is a vector, then ,A is
equivalent to A; if A is a scalar, then ,A is a vector of length l.
The dyadic function p reshapes its right argument to the shape
specified by its left argument. If M+DpV, then M is an array of
dimension D whose elements are the elements of V. For example,
2 3p 123456 is the matrix
If n, the total number of elements required in the array DpV, is
equal to the dimension of the vector V, then the ravel of DpV is equal
to V. If N is less than pv , then only the first N elements of V are
used; if N is greater than pV , then the elements of V are repeated
cyclically. For example:
2 3pl 2
2 1
1 2
3 3pl
10
10
1
More generally, if A is any array, then DpA is equivalent to Dp ,A-
For example:
A
12 3
4 5 6
3 5pA
2 3 4
12 3
6 12
The expressions OpX and o 3pX and o OpJ are all valid; any one or
more of the axes of an array may have zero length. Such an array is
called an empty array . If D is an empty vector, then DpA is a scalar.
32 APL Language
Reverse and Rotate
The monadic function reverse is denoted by <|> ; if x is a vector and
K+QX , then k is equal to x except that the items appear in reverse
order. The axis operator applies to reversal and determines the axis
along which the vectors are to be reversed. For example:
A $[114 <t>[2lU
123 H56 321
456 123 6 5 4
The expression $A denotes reversal along the last coordinate of A ,
and ba denotes reversal along the first coordinate. For example, if A
is of rank 3, then $A is equivalent to <$[3]A , and qA is equivalent to
4>[l]i4.
The dyadic function rotate is also denoted by (J> . If if is a scalar
or one-element vector and x is a vector, then Ki?X is a cyclic rotation
of x defined as follows: K$X is equal to z[i+ ( p j) | ~l+K+\pXl . For
example, if x<-2 3 5 7 11, then 2$X is equal to 5 7 11 2 3 , and ~2$X is
equal to 7 11 2 3 5. In o-origin indexing, the definition for K$X
becomes XL (p X) \ K+ ipZ] .
If the rank of x exceeds 1 , then the coordinate J along which
roatation is to be performed may be specified by the axis operator in
the form z-*-K$LJlX. Moreover, the shape of K must equal the remaining
dimensions of x, and each vector along the Jth axis of x is rotated as
specified by the corresponding element of K. A scalar K is extended in
the usual manner.
For example, if p x is 3 4 and J is 2 , then the shape of K must be
3 and Z[J;3 is equal to £[J]4>J[I; ] . If J is 1 , then pK must be 4, and
Z[;J] is equal to X[I]<t>X[ ;T] . For example:
M
1 2 3<)>[1]M
1 2 3C|)[2]W
1
2
3
4
1
6
11 4
2
3 4 1
5
6
7
8
5
10
3 8
7
8 5 6
9
10
11
12
9
2
7 12
12
9 10 11
The expression K&X denotes rotation along the first axis of x.
Catenate and Laminate
Catenate , denoted by a comma, chains vectors (or scalars) together
to form a vector. For example:
X+2 3 5 7 11
X,X
2357 11 2357 11
In general, the dimension of x ,Y is equal to the total number of
elements in x and y. A numeric vector cannot be catenated with a
character vector.
The axis operator applies to catenate and determines the axis
along which vectors are to be catenated. For example;
M-i-3
3p
'ABCDEFGHI<
M
M,
[1]M
M.
,[2]M
M,M
ABC
ABC
ABCABC
ABCABC
DEF
DEF
DEFDEF
DEFDEF
GHI
GHI
ABC
DEF
GHI
Section
3:
GHIGHI
Primative
Funct
GHIGHI
ions and Op
33
Two arrays are conformable for catenate along axis I if all other
elements of their shapes agree. Moreover, two arrays may be catenated
along axis I if they differ in rank by 1 and if the shape of the array
of lower rank equals the shape of the array of higher rank after
dropping item I from it. For example:
V^'PQR'
M.IX1V M,L21V M.V
ABC ABCP ABCP
DEF DEFQ DEFQ
GHI GHIR GHIR
PQR
A scalar argument of catenate will be replicated to form a vector as
required. For example:
<7,(C,[1] W.C1] C) ,C
mm®
EWBCS
[SOfi'FS
BCfflffl
mmm
Laminate joins two arrays of identical rank and shape along a new
axis. The position of the new axis relative to the existing axes is
indicated by a fractional axis number. For example, if the new axis is
to be inserted between the existing axes 1 and 2 , the axis number must
have a value between 1 and 2 . If the new axis is to be inserted ahead
of the present first axis, the axis number must be between and 1 (or,
if zero-origin indexing is used, between ~1 and 0). Similarly, if the
new axis is to be after the last of the present axes, the axis number
must exceed the index of the present last axis by a fraction between
and 1 .
The result of lamination has rank 1 greater than the rank of the
arguments, and has the same shape except for the interpolation of the
new axis, along which it has length 2. The comma, which normally
denotes catenation, followed by an axis operator associated with a
non-integral index produces lamination. For example:
5] 217
W-<-3
3p
'ABCDEFGHI'
2N-3
3p
'123456789'
M
M,l
ABC
ABC
DEF
DEF
GUI
N
GHI
123
123
456
456
789
789
M.
,[1.
.5] 217
M,L2.51N
ABC
Al
123
B2
C3
DEF
456
04
2?5
GHI
F6
789
Gl
H8
J9
The shapes of the foregoing laminations are 2 3 3 and 3 2 3 and
3 3 2; the position of the 2 shows the point of new axis insertion in
each case.
34 APL Language
A scalar argument of laminate is extended as required
example :
For
B-<-2 2p'1234'
B,[2.5]'+'
1 +
2 +
3 +
4 +
,B,[2.5]»+»
1+2+3+4+
Transpose
The expression 2 1W yields the transpose of the matrix M; that
is, if i?x-2 1§M , then each element R[I;Jl is equal to MM ill. For
example : '
M
2 ltytf
1
2 3
4
1
5
9
5
6 7
8
2
6
10
9
10 11
12
3
7
11
4
8
12
If p xs any permutation of the indices of the axes of an array A
then ptstA is an array similar to A except that the axes are permuted:
the jth axis becomes the p[T]th axis of the result. Hence, if E<-p<siA ,
then (pi?)[P] is equal to pA . For example:
A+2 3 5 7pi210
PA
2 3 5 7
P-e-2 3 4 1
pPM
7 2 3 5
More generally, q§a is a valid expression if Q is any vector equal
in length to the rank of A which is "complete" in the sense that if its
items include any integer N they also include all positive integers less
than n. For example, ±f ppA is 3, then 112 and 2 11 and 111 are
suitable values for q but 1 3 1 is not. Just as for the case P§A where
P is a permutation, the Jth axis becomes the Q[l]th axis of Q§A.
However, in this case two or more of the axes of .4 may map into a single
axis of the result, thus producing a diagonal section of A as
illustrated below:
4-e-3 3pl9 S-t-3 5pil5
A B
12 3 12 3 4
4 5 6 6 7 8 9
7 8 9 11 12 13 14
1 1§A 1 l(sjB
15 9 1 7 13
5
10
15
The monadic transpose ts>A reverses the order of the axes of its
argument. Formally, ts>A is equivalent to (<t>ippd)*}d. In particular, for
a matrix a this reduces to 2 1 ^ and is what is commonly called the
transpose of a matrix.
Section 3: Primative Functions and Operators 35
SELECTION FUNCTIONS
The selection functions are all dyadic; one of the arguments may
be. an array of any type, and the other, which will be referred to as the
selector because it serves to specify the selection to be made, must be
numeric, and, in the case of expand and compress , is further restricted
to boolean.
Take a nd Drop
If s is a non-negative scalar integer and V is a vector, then S + V
is a vector of dimension s obtained by taking the first S elements of V
followed (if s>pV) by zeros if v is numeric and by spaces if it is not.
For example:
3+2 3 5 7 7 + 2 3 5 7
235 2357000
3+MBCDE 1 a+'ABCDE' ) ,*@'
ABC ABODE @
If 5 is a negative integer, then s + V takes elements as above but
takes the last elements of v and fills as needed on the left. For
example :
3+2 3 5 7
3 5 7
7+2 3 5 7
2 3 5 7
If a is any array, then w+A is valid only if the vector W has one
element for each axis of a, and will determines what is to be taken
along the jth axis, of 4. For example:
A+3 4pil2
A
12 3 4
5 6 7 8
9 10 11 12
12 3
5 6 7
2 3+4
"3+4
2 3 4
6 7 8
6
10
'2 6+4
7
11
12
The function drop U) is defined analogously, except that the
indicated number of elements are dropped rather than taken. For
example, "1 n-A is the same, matrix as the result of 2 ~3+4 displayed in
the preceding paragraph. If the number of elements to be dropped along
any axis equals or exceeds the length of that axis, the resulting shape
has a zero length for the axis.
The rank of the result of the take and drop functions is the same
as the rank of the right argument.
Compress and Expand
The expression u/X denotes compression of X by U. If U is a
boolean vector and x is a vector of the same dimension, then U/X
produces a vector of +/y elements chosen from those elements of X
corresponding to non-zero elements of u. For example, if X+2 3 5 7 11
and u+1 110 then u/X is 2 5 7 and (~U)/X is 3 11.
C+'THIS IS AN EXAMPLE'
D+-C*' '
Cl+D/C
CI
THISISANEXAMPLE
To be conformable, the dimensions of the arguments must agree,
except that a scalar argument is extended. Hence l/X is equal to X and
q/X is an empty vector.
36 APL Language
Expansion is the converse of compression and is denoted by u\X .
If I<-U\X, then u/Y is equal to x and (~y)/j is an array of zeros or
spaces according as x is numeric or character. In other words, u\X
expands x to the format indicated by the ones in u and fills in zeros or
spaces. To be conformable, +/u must equal p x.
D\C1
THIS IS AN EXAMPLE
The axis operator applies to both compress and expand and
determines the axis along which they apply. If the axis operator is
omitted, the last axis is used. The symbols / and \ also denote
compression, but when used without an axis operator apply along the
first axis.
Q+3 hp'ABCDEFGHIJKL'
ABCD
EFGH
IJKL
ABCD
EFGH
IJKL
ABCD
EFGH
IJKL
110 1\[1]<
1 1 1\(
BC
FG
JK
ABCD
IJKL
ABCD
IJKL
Oil 0/<3
1 1/lllQ
1 1/Q
The rank of the result of compress or expand equals the rank of
the right argument.
Indexing
Indexing may be either p- origin or l- origin as discussed in
Section 2. The following discussion assumes l-origin. If x is a vector
and I is a scalar, then XLH denotes the jth element of x. For example,
if X+2 3 5 7 11 then x[2] is 3.
If the index j is a vector, then xlH is the vector obtained by
selecting from x the elements indicated by successive components of j.
For example, xil 3 5] is 2 5 11 and XL 5 4 3 2 l] is n 7 5 3 2. If the
elements of j do not belong to the set of indices of x, then the
expression x[r] evokes an index error report.
In general pXZH equals p j. In particular, if j is a scalar, then
XLH is a scalar, and if j is a matrix then x[j] is a matrix. For
example:
A+-<ABCDEFG'
I«-4 3p3 1 t 2 1 1 1 1 2 1 1 1
I
3
1
4
CAD
2
1
4
BAD
4
1
2
DAB
4
1
4
DAD
ALII
If M is a matrix, then m is indexed by a two-part list of the form
1% J where 1 selects the row (or rows) and j selects the column (or
Section 3: Primative Functions and Operators 37
columns) . For example:
M M[2;3] Mil 3;2 3 4]
1234 7 234
5 6 7 8 10 11 12
9 10 11 12
In general, pMH;J~\ is equal to (pI),pJ. Hence if J and J are
both vectors, then Mil; J~\ is a matrix; if both I and J are scalars,
MiI;J'] is a scalar; if I is a vector and J is a scalar (or vice versa) ,
MiI;J'} is a vector. The indices are not limited to vectors, but may be
of higher rank. For example, if I is a 3 by 4 matrix, and J is a vector
of dimension 6, then Mil; J] is of dimension 3 4 6, and MiJ;I~\ is of
dimension 6 3 4. In particular, if T and P and Q are matrices, and if
R+TIP
TlPlI
], then R is an array of rank 4 and PC J ;J;K;L] is equal to
iJliQlK-.Ll .
The form Mil;'] indicates that all columns are selected, and the
form Mi;J~\ indicates that all rows are selected. For example, Mi 2 ; ] is
5 6 7 8 and Mi ; 2 1] is the matrix with rows 2 1 and 6 5 and 10 9.
The following example illustrates the use of a matrix indexing a
matrix to obtain a three-dimensional array:
A/f-2 4p 31421441
M Mi ;M]
3142 4321
1441 3223
4 114
1111
An indexed variable may appear to the left of a specification
arrow if (1) the expression is executable in the environment and (2)
denoting the values of the expression on the left and right by L and R,
then l=x/pR or (l*p£)/p£ must equal ( l*pP)/pP. For example:
X+2 3 5 7 11
XI 1 3]-*-6 8
X
6 3 8 7 11
SELECTOR GENERATORS
All of the functions in this group have integer results which,
although they are commonly useful as the selector argument in selection
functions, are often used in other ways as well. For example, the grade
function i is commonly used to produce indices needed to reorder a
vector into ascending order (as in XiiX]) , but may also be used in the
treatment of permutations as the inverse function, that is, ^P yields
the permutation inverse to P. Similarly, \N generates a vector of N
successive indices, but . lxiff generates a grid of values with an
interval of . 1 .
Index Generator and Index of
The index generator i applies to a non-negative scalar integer N
to produce a vector of length N containing the first N integers in
order, beginning with the value of the index origin DIO . For example,
r5 yields 12 3 4 5 (in 1-origin) or 1 2 3 4 (in 0-origin) , and iO
yields an empty vector. A one-element vector argument is treated as a
scalar.
If V is a vector and S is a scalar, then V\S yields the index (in
38 APL Language
the origin in force) of the earliest occurence of s in v, that is, the
index of S in V. If S differs from all items of v, then V\S yields the
first index outside the range of V, that is, UlO+pV.
If S is any array, then V\S yields an array of the shape of s,
each item being determined as the index in v of the corresponding item
of S. For example:
A+'ABCDEFGHIJKLMNOPQRSTUVWXYZ '
J+A I ' HEAD
CHIEF '
M«-2 5p'
J
M
8 5 1 4 27 3 8 9
I 5
6
HEAD
AU1
CHIEF
HEAD CHIEF
A\M
AlbJl
8 5 1 4 27
FEIHC DAEH
3 8 9 5 6
A\ 'VAB3 1
22 1 18 28
Membership
HEAD CHIEF'
The function XeY yields a boolean array of the same shape as x.
Any particular element of XeY has the value l if the corresponding
element of x belongs to Y, that is, if it occurs as some element of y.
For example, (i7)e3 5 is equal to 0010100 and » ABCDEFGH' e' COFFEE'
equals 00101100. The right argument y may be of any rank.
The selector argument of compression is commonly provided by
application of the membership function, alone or in combination with the
scalar boolean and relational functions.
Grade Functions
The grade up function &v grades the items of vector v in ascending
order, that is, it yields a result of the same dimension as v whose
first item is the index (in the origin in force) of the smallest item of
V, whose second item is the index of the next smallest item, and so on.
Consequently, the expression VlbVl yields the elements of V in ascending
order. For example, if 7*8 3 7 5, then |Hs 2 t 3 1 and VLiVl is
3 5 7 8.
If the items of V are not all distinct, the ranking among any set
of equal elements is determined by their position. For example,
A4 3 1 3 4 2 yields 3 6 2 4 15.
The grade down function yv grades .the items of V in descending
order; among equal elements the ranking is determined by position just
as for grade up. Consequently, yv equals the reversal of |7 only if the
items of V are distinct. For example:
A+7 2 5 11 3
iA
2 5 3 14
4 13 5 2
B«-4 3 13 4 2
iB
3 6 2 4 15
15 2 4 6 3
The grade functions apply only to vectors,
Deal
The function M?N produces a vector of length M obtained by making
M (pseudo-) random selections, without replacement, from the population
Section 3 : Primative Functions and Operators
39
\N. Both arguments are limited to scalars or one-element vectors. Each
selection is made by appropriate application of the scheme described for
the function roll .
The expression N?N yields a random permutation of the items of \N .
The expression p[A/? p p] selects M distinct elements from the population
defined by the items of a vector p . For example :
GBD
P*-<ABCDEFGH*
PL3?pP]
P[(pP)?pP]
CEADHFGB
NUMERICAL FUNCTIONS
The numerical mixed functions apply only to numerical arguments
and produce numerical results.
Matrix Inverse and Matrix Divide
The domino (g) represents two functions which are useful in a
variety of problems including the solution of systems of linear
equations, determining the projection of a vector on the subspace
spanned by the columns of a matrix, and determining the coefficients of
a polynomial which best fits a set of points in the least-squares sense.
When applied to a non-singular matrix A the expression SU (matrix
inverse) yields the inverse of A, and the expression X*-BM (matrix
divide) yields a value of X which satisfies the relation a/ ,b=A+ .xX and
is therefore the solution of the system of linear equations
conventionally represented as Ax=b:
A+i
U)«
. >
u
A
m
A+,
. x@/l
1
1
1
1 1
1
1
1
1 1
1
1 1
1
1 1
1 1
1
1
1
B<-1 3 6 10
x+Bm
B X
3 6 10 12 3 4
A+.xX
1 3 6 10
(1/1) + . *B
12 3 4
Ct-H 2pl 2 3 5 6 9 10 14
■cm
10 14
A+.
1 2
3 5
6 9
10 14
xY
(BA)+.xC
The final example above shows that if the left argument is a matrix C,
then c®A yields a solution of the system of equations for each column of
C
If A is non-singular and if j is an identity matrix of the same
dimension, then the matrix inverse SU is equivalent to the matrix divide
TIE4. More generally, for any matrix p, the expression @P is equivalent
to the expression ( ( iP) o . = iP)ip, where R is the number of rows in P.
The domino functions apply more generally to singular and
non-square matrices, and to vectors and scalars; any argument of rank
greater than 2 is rejected (RANK ERROR) . For matrix arguments A and B
HO APL Language
the expression x+B%A is executed only if
1. A and B have the same number of rows, and
2. the columns of A are linearly independent.
If the _ expression X+BM is executable, then pX is equal to (i + Pi i) i + p b
and X xs determined so as to minimize the value of the expression'
+ / AB-A+.xX)*2.
The domino functions apply to vector and scalar arguments as
follows: except that the shape of the result is determined as specified
above, a vector is treated as a one-column matrix and a scalar is
treated as a one-by-one matrix. The rationale for this interpretation
of a vector as a one-column (rather than one-row) matrix is that the
right argument is treated geometrically (as will be seen in a subsequent
example) as defining a space spanned by its column vectors, and the left
argument was seen (in an earlier example) to be treated so as to yield a
solution for each of its column vectors. Indeed, a one-rowed matrix
right argument (unless 1-by-l) would be rejected under condition 2
above .
In the case of scalar arguments X and Y, the expression @J is
equivalent to *y and, except that it yields a domain error for the case
0@0, the expression X®Y is equivalent to X*Y .
The use of H for a singular right argument can be illustrated as
follows: if x is a vector and Y+F X, then the expression Ym°.*Q,iD
yields the coefficients of the polynomial of degree D which best fits
(in the least squares sense) the function F at the points X.
The definition of BMA has certain useful geometric
interpretations. If B is a vector and A is a matrix, then saying that
+ /(.B-A + .*Bm)*2 is a minimum is equivalent to saying that the length of
the vector B-A+.xBm is a minimum. But A+.xBftA is a point in the space
spanned by the column vectors of A and is therefore the point in this
space which is closest to B. In other words, P+A + . xBBA is the
projection of B on the space spanned by the columns of A. Moreover, the
vector B-P must be normal to every vector in the space; in particular,
(B-P)+ .xA is a zero vector.
If A and B are single-column matrices, then BEA is a l by 1 matrix
and A+.xBM is equivalent to AxS , where S is the scalar "pBS^. if A
and B are vectors, then BM is a scalar and the projection of B on A is
therefore given by the simpler expression AxBm. For example:
4«-4.5 1.7
Bf-2 5
P+AxB%A
P
3.403197926 1.28565255
N+B-P
N
1.4-03197926 3.71434745
N+.xA
2.442490654£' - 15
Similar analysis shows that if A is a vector then m is a vector
in the direction of A; that is, m is equal to SxA for some scalar S.
Moreover, A+ .x$a is equal to l . In other words, m is the "image" of
Section 3: Primative Functions and Operators 41
the vector A obtained by inversion in the unit circle (or sphere)
Decode and Encode
For vectors R and X , the decode (or base-value ) function R±X
yields the value of the vector X evaluated in a number system with
radices i?[l], RL 2] RtpRl. For example, if B+21 60 60 and x+l 2 3 is
a vector of elapsed time in hours, minutes, and seconds, then RiX has
the value 3723, and is the corresponding elapsed time in seconds.
Similarly, 10 10 10 10 l 1 7 7 6 is equal to 1776, and 222x101 is
equal to 5. Formally, RiX is equal to + /J/xX, where W is the weighting
vector determined as follows: WLpWl is equal to 1 and WLJ-ll is equal
to RLIlxWLIl. For example, if R is 24 60 60, then W is 3600 60 1.
Scalar (or one-element vector) arguments are treated as usual.
For example, 10 i 1 7 7 6 yields 1776. The arguments are not restricted
to integers; for example, if X is a scalar then XiC is the value of the
polynomial with coefficients C, arranged in descending order on the
powers of X .
The decode function is extended to arrays in the manner of the
inner product operator: each of the radix vectors along the last axis
of the first argument is applied to each of the vectors along the first
axis of the second argument. There is one difference; if either of
these distinguished axes is of length 1 , it will be extended (by
replication of the element) as necessary to match the length of the
other argument. Except for this different treatment of unit axes, the
shape of the result of Ax.B is determined like the shape of the inner
product, namely (~l + p4) ,1 + pB .
The encode (or representation ) function RiX is, for certain
arguments, inverse to the decode function. For example:
i?«-10 10 10 10
i?±l 7 7 6
1776
i?Tl776
17 7 6
More generally, Rl{RjX) equals {*/R)\X rather than X. For example:
10 10 10 10T123 10 10 10T123
12 3 12 3
10 10T123 10T123
2 3
More precisely, the definition of the encode function is based on
the definition of the residue function; for a vector left argument and
scalar right argument, decode is equivalent to the E whose canonical
representation is shown at the left below:
Z+A E B 2 2 2T13
Z«-0xA 10 1
I+pA "2 "2 "2T13
L:-KI=0)/0 1 "1 1
ZZI1+ALI1\B 2 2T13
■+U[I] = 0)/0 6 1
B+(B-ZLI1)*AII1 2 2 2T 13
I-t-I-1 11
+£, "2 2 2T13
11
42 APL Language
The basic definition of RtX concerns a vector R and a scalar X ,
and produces a result of the shape of if . It is extended to arrays as
follows: each radix vector along the first axis of R is applied to
obtain the representation of each item of X , the resulting
representations being arrayed along the first axis of the result. For
example :
10 10 10T215 4-86 72 219 3 i?+-10 10 10,[1.5]8 8 8
2 4 2 R
18 7 10 10 8
5 6 2 9 3 1Q 8
10 8
i?Tl23
1 1
2 7
3 3
The expression for the shape of the result of RjX is the same as
for the shape of the outer product, namely ( P R), p x.
DATA TRANSFORMATIONS
Of the two functions in this class, the format is a true type
transformation, being designed to produce a character array which
represents the data in its numeric argument. Over a certain class of
arguments the execute function is inverse to the format and is therefore
considered as a type transformation as well, although its applicability
is, in fact, much broader.
Execute and Format
Any character vector or scalar can be regarded as a representation
of an APL statement (which may or may not be well-formed) . The monadic
function denoted by a. takes as its argument a character vector or scalar
and evaluates or executes the APL statement it represents. When applied
to a character array that might be construed as a system command or the
opening of function definition, an error will necessarily result when
evaluation is attempted, because neither of these is a well- formed APL
statement.
The execute function may appear anywhere in a statement, but it
will successfully evaluate only valid (complete) expressions, and its
result must be at least syntactically acceptable to its context. Thus,
execute applied to a vector that is empty, contains only spaces, or
starts with "-*• (branch symbol) or n (comment symbol) produces no explicit
result and therefore can be used only on the extreme left. For example:
4' •
VALUE ERROR
Z«-±' '
A
The domain of ± is any character array of rank less than two, and
RANK and DOMAIN errors are reported in the usual way:
C«-'3 4'
+/aC ±3 4
7 DOMAIN ERROR
11 3pC 43 4-
RANK ERROR a
1 1 3 pC
A
Section 3: Primative Functions and Operators 43
An error can also occur in the attempted execution of the APL
expression represented by the argument of i ; such an indirect error is
reported by the error type prefaced by the symbol 4 and followed by the
character string and the caret marking the point of difficulty. For
example :
a. DOMAIN ERROR
4*0
A
1 * ) WSID '
4 VALUE ERROR
)WSID
A
The symbol <r denotes two format functions which convert numerical
arrays to character arrays. There are several significant uses of these
functions in addition to the obvious one for composing tabular output.
For example, the use of format is complementary to the use of execute in
treating bulk input and output, and in the management of combined
alphabetic and numeric data.
The monadic format function produces a character array which will
display identically to the display normally produced by its argument,
but makes this character array explicitly available. For example:
M«-2=?4 4p2
R+lM
M R 7?[;2xi4]
0101 0101 0101
0011 0011 0011
1011 1011 1011
0011 0011 0011
pM pi?
4 4 4 8
p<T2 5 Xf-34
3 X THE VALUE OF X IS ' ,lX
TEE VALUE OF X IS 34
1
ABCD
A/,i? = Ti?
▼ MBC7? 1
The format function applied to a character array yields the
array unchanged, as illustrated by the last two examples above. For a
numerical array, the shape of the result is the same as the shape of the
argument except for the required expansion along the last coordinate,
each number going, in general, to several characters. The format of a
scalar number is always a vector.
The printing normally produced by APL systems may vary slightly
from system to system, but the result produced by the monadic format
will have no final column of all spaces, and no initial spaces in the
case of a vector or scalar argument.
The dyadic format function accepts only numerical arrays as its
right argument, and uses variations in the left argument to provide
progressively more detailed control over the result. Thus, for EvA , the
argument F may be a single number, a pair of numbers, or a vector of
length 2x~l-M,p J 4.
In general, a pair of numbers is used to control the result: the
first determines the total width of a number field, and the second sets
the precision . For decimal form the precision is specified as the
HH APL Language
number of digits to the right of the decimal point, and for scaled form
it is specified as the number of digits in the multiplier. The form to
be used is determined by the sign of the precision indicator, negative
numbers indicating scaled form. Thus: negative
3 2
pO+A
12 . 34
_0
0.26
i?«-9 2fA
S+9 ~2<fA
34.567
12
"123. 45
3 24
pD«-12 3fA
12.340 "34.567
.000 12.000
.260 "123.450
12.34
_.00
.26
3 18
pQf-5
1.2501
_0 .0500
2.65 01
3 18
34.57
12 .00
"123. 45
5501
2501
2502
pD*-6 Of A
12 "35
12
123
3 12
pD«-7 ~1*A
1501 ~3EQ1
0E00 1E01
~35~01 ~±E02
3 14
If the width indicator of the control pair is zero, a field width
is chosen such that at least one space will be left between adjacent
numbers. If only a single control number is used, it is treated like a
number pair with a width indicator of zero:
p0-*-2vA
12.34 "34.57
_.00 12.00
.26 "123.45
3 16
pQ-f-0 2vA
12.34 "34.57
.00 12.00
".26 "123.45
3 16
pD+-~2<M
1.2501 "3.5501
0.050 1.2501
~2.65~01 "1.2502
3 18
pCH-0 ~2VA
1.2501 ~3. 5E01
0.0500 1.2501
~2.65 01 "1.2502
3 18
Each column of an array can be individually composed by a left
argument that has a control pair for each:
pD«-0 2 2wA
12.34 "34.57
.00 12.00
".26 "123.45
3 15
pQ-<-6 2 12 ~3Ti4
12.34 "3.46S01
.00 1.20501
".26 "1.23502
3 18
pQ«-8_3 2JA
12.340 34.57
.000 12.00
".260 "123.45
3 16
pQf-8 ~2Ti4
12 "3.5501
1.2501
"1.2502
3 17
6283304050 12 4t ,A
12.34 34.567 12 "123.4500
The format function applied to an array of rank greater than two
applies to each of the planes defined by the last two axes. For
example :
Section 3: Primative Functions and Operators
45
L«-2 = ?2 2 5p2
L 4 1TL
110 1 1.01.0 0.00.01.0
1110 1 i.o 1-0 1 -° °-° 1 -°
10 10 1.00.00.01.00.0
ooooo 0.00.00.00.00.0
Tabular displays incorporating row and column headings, or other
information between columns or rows, are easily configured using the
format function together with catenation. For example:
ROWHEADS*-k Zq'JANAPRJULOCT'
YEARS+71-HS
TABLE*-. 001x~4£5+?4 5p8£5
(• » ,il^ROWHEADS), (.2^9 OvYEARS) , [ 1 ] 9 If TABLE
72 73 7 4 75 _76
JAN 318.13 "351.55 _ 3.62 "144.77 4.82
APR "327.41 "341.00 "92.69 331.05 28.44
JUL "359.93 216.16 _299..71 150.77 103.64
OCT 180.33 310.86 154.94 10.62 276.79
There are obvious restrictions on the left argument of format,
since the width of a field must be large enough to hold the requested
form, and if the specified width is inadequate the result will be a
DOMAIN error. However, the width need not provide open spaces between
adjacent numbers. For example, boolean arrays can be tightly packed:
1 0?2=?4 4p2
1001
0000
1101
0111
The following formal characteristics of the format function need
not concern the general user, but may be of interest in certain
applications :
The least width needed for a column of numbers C with precision P is
RMv/f?<o) + (~PeO ~l) + ( |P) + (4,("/0,(i?*0) + L10®|i?+i?=0)[l+P2:0], where S is
the rounded value of C given by i?«-(L . 5 + C*io* I P)*10* I P.
The expressions (MvA) ,N*B and (M,N)vA,B are equivalent if M and N
are full control vectors, that is, if ( ( P A/)=2x"l+ P A ) A(ptf)=2x 1+pB.
If 2 = pA7, then (MjA) ,MjB and MvA.B are equivalent.
46 APL Language
SECTION 4 : SYSTEM FUNCTIONS AND SYSTEM VARIABLES
Although the primitive functions of APL deal only with abstract
objects (arrays of numbers and characters), it is often desirable to
bring the power of the language to bear on the management of the
concrete resources or the environment of the system in which APL
operates. This can be done within the language by identifying certain
variables as elements of the interface between APL and its host system,
and using these variables for communication between them. While still
abstract objects to APL, the values of such system variables may have
any required concrete significance to the host system.
In principle all necessary interaction between APL and its
environment could be managed by use of a complete set of system
variables, but there are situations where it is more convenient, or
otherwise more desirable, to use functions based on the use of system
variables which may not themselves be made explicitly available. Such
functions are called, by analogy, system functions .
System variables and system functions are denoted by distinguished
names that begin with a quad (Q) . The use of such names is reserved for
the system and cannot be applied to user-defined objects. They cannot
be copied, grouped, or erased; those that denote system variables can
appear in function headers, but only to be localized (see Section 6) .
Within APL statements, distinguished names are subject to all the normal
rules of syntax.
SYSTEM FUNCTIONS
Like the primitive abstract functions of APL, the system functions
are available throughout the system, and can be used in defined
functions. They are monadic or dyadic, as appropriate, and have
explicit results. In most cases they also have implicit results, in
that their execution causes a change in the environment. The explicit
result always indicates the status of the environment relevant to the
possible implicit result.
Altogether, 13 system functions are provided. Six of these are
concerned with the management of the shared-variable facility and are
described in Section 5. The other seven are given in Figure 4.1, and
are described here.
CANONICAL REPRESENTATION
The canonical representation of a defined function as defined in
Section 6 is obtained as a result of applying the system function OCR to
the character array representing the name of the function. Applied to
any argument which does not represent the name of an unlocked defined
function it yields a matrix of dimension o by o. Possible error reports
for OCR are 'RANK error if the argument is not a vector or a scalar, or
DOMAIN error if the argument is not a character array. The use of OCR
is further described in Section 6.
FUNCTION ESTABLISHMENT
The definition of a function can be established or fixed by
applying the system function OFX to its character representation. The
function OFX produces as an explicit result the array of characters
which represents the name of the function being fixed, while replacing
any existing definition of a function with the same name.
Section 4: System Functions and System Variables 47
FUNCTION
UCR A
UFX M
OEX A
DNL N
REQUIREMENTS
RANK LENGTH
l>pp4
2 = ppM
2>ppA
l>ppff
A DNL N
DNC A
DDL S
lappiV
2£ppM
lappS
lip. 5
DOMAIN
Array of
characters.
Matrix of
characters.
Array of
characters.
A/Nel 2 3
A/flel 2 3
Elements of
A must be
alphabetic.
Array of
characters.
EFFECT ON ENVIRONMENT
None.
Fix (establish) definition of
the function represented by M,
unless its name is already in
use for an object other than
function which is not halted.
Expunge (erase) objects named
by rows of A, except groups,
labels, or halted functions.
None.
None.
None.
Numeric
value.
None, but requires S seconds
to complete.
EXPLICIT RESULT
Canonical representation of
object named by A. The re-
sult for anything other than
an unlocked defined function
is of size 0.
A vector representing the
name of the function estab-
lished, or the scalar row
index of the fault which
prevented establishment.
A boolean vector whose Ith
element is 1 if the Ith name
is now free.
A matrix of rows (in acci-
dental order) representing
names of designated kinds in
the dynamic environment:
1, 2, 3 for labels, varia-
bles, functions.
As for the monadic form,
except that only names be-
ginning with letters in A
will be included.
A vector giving the usage of
the name in each row of A :
name is available
1 label
2 variable
3 function
4 other
Scalar value of actual
delay.
Figure 4.1: System Functions
An expression of the form DFX M will establish a function if both the
following conditions are met:
1. M is a valid representation of a function. Any matrix which
differs from a canonical matrix only in the addition of
non- significant spaces (other than rows consisting of spaces
only) is a valid representation.
2. The name of the function to be established does not conflict
with an existing' use of the name for a halted function (defined
in Section 7) or for a label, group, or variable.
If the expression fails to establish a function then no change occurs in
the workspace and the expression returns a scalar index of the row in
the matrix argument where the fault was found. If the argument of UFX
is not a matrix a RANK error will be reported, and if it is not a
character array a DOMAIN error will result.
EXPUNGE
Certain name conflicts can be avoided by using the expunge
function OEX to dynamically eliminate an existing use of a name. Thus
OEX •Pejf?' will erase the object PQR unless it is a label, a group, or a
48 APL Language
halted function. The function returns an explicit result of l if the
name is now unencumbered, and a result of o if it is not, or if the
argument does not represent a well- formed name. The expunge function
applies to a matrix of names and then produces a logical vector result.
QEX will report a RANK error if its argument is of higher rank than a
matrix, or a DOMAIN error if the argument is not a character array. A
single name may also be presented as a vector or scalar.
NAME LIST
The dyadic function QNL yields a character matrix, each row of
which represents the name of an object in the dynamic environment. The
right argument is an integer scalar or vector which determines the class
of names produced as follows: 1, 2, and 3 respectively invoke the names
of labels, variables, and functions. The left argument is a scalar or
vector of alphabetic characters which restricts the names produced to
those with an initial letter occurring in the argument. The ordering of
the rows of the result is fortuitous .
The monadic function QNL behaves analogously with no restriction
on initial letters. For example, QNL 2 produces a matrix of all
variable names, and either of QNL 2 3 or QNL 3 2 produces a matrix of
all variable and function names.
The uses of 'QNL include the following:
In conjunction with QEX, all the objects of a certain class can
be dynamically erased; or a function can be readily defined that
will clear a workspace of all but a preselected set of objects.
In conjunction with QCR , functions can be written to
automatically display the definitions of all or certain
functions in the workspace, or to analyze the interactions among
functions and variables.
The dyadic form of QNL can be used as a convenient guide in the
choice of names while designing or experimenting with a
workspace.
NAME CLASSIFICATION
The monadic function QNC accepts a matrix of characters and
returns a numerical indication of the class of the name represented by
each row of the argument. A single name may also be presented as a
vector or scalar.
The result of QNL is a suitable argument for QNC, but other
character arrays may also be used, in which case the possible results
are integers ranging from to 4. The significance of l, 2, and 3 are
as for QNL} a result of signifies that the corresponding name is
available for any use; a result of u signifies that the argument is not
available for use as a name. The latter case may arise because the name
is in use for denoting a group, or because the argument is a
distinguished name or not a valid name at all.
DELAY
The delay function, denoted by QDL , evokes a pause in the
execution of the statement in which it appears. The argument of the
function determines the duration of the pause, in seconds, but the
accuracy is limited by possible contending demands on the system at the
moment of release. Moreover, the delay can be aborted by a strong
Section 4: System Functions and System Variables 49
interrupt. The explicit result of the delay function is a scalar value
equal to the actual delay. If the argument of DDL is not a scalar with
a numerical value, a RANK or DOMAIN error will be reported.
Generally speaking, the delay function uses only a negligible
amount of computer time (as opposed to connect time) . It can therefore
be used freely in situations where repeated tests may be required at
intervals to determine whether an expected event has taken place. This
is useful in work with shared variables (as in the function OFFER given
as an example in Section 5) , as well as in certain kinds of interactions
between users and programs.
SYSTEM VARIABLES
System variables are instances of shared variables , which are
treated in Section 5. The characteristics of shared variables that are
most significant here are these:
1. If a variable is shared between two processors, the value of the
variable when used by "one of them may well be different from
what that processor last specified, and
2. each processor is free to use or not use a value specified by
the other, according to its own internal workings.
System variables are shared between a workspace and the APL
processor. Sharing takes place automatically each time a workspace is
activated and, when a system variable is localized in a function, each
time the function is used.
The system variables are listed in Figure 4.2, which gives their
significance and use. Two classes can be discerned:
1. Comparison tolerance, index origin, latent expression, random
link, printing precision, and printing width. In these cases
the value specified by the user (or available in a clear
workspace) is used by the APL processor during the execution of
operations to which they relate. If this value is
inappropriate, or if no value has been specified after
localization, an IMPLICIT error will be evoked at the time of
execution.
2. Account information, atomic vector, line counter, time stamp,
terminal type, user load, and work area. In these cases
localization or setting by the user are immaterial. The APL
processor will always reset the variable before it can be used
again.
The APL statement represented by the latent expression is
automatically executed whenever the workspace is activated. Formally,
DLX is used as an argument to the execute function (*QLX) , and any error
message will be appropriate to the use of that function.
Common uses of the latent expression include the form QLX+'G* used
to invoke an arbitrary function G, the form:
QLX+ " ' FOR NEW FEATURES IN THIS WS ENTER: NEW"
used to print a message upon activation of the workspace, and the form
DLX-*- , -*ULC t used to automatically restart a suspended function. The
variable DLX may also be localized within a function and respecified
therein to furnish a different latent expression when the function is
50 APL Language
NAME
PURPOSE
VALUE IN
CLEAR WS
MEANINGFUL
RANGE
uct
Comparison tolerance: used in monadic f L, dyadic << = £>*£ i
1£~13
0-1
DIO
Index origin: used in indexing and in ? i i H QP-^
1
1
ULX
Latent expression executed on activation of workspace
i t
characters
DPP
Printing precision: affects numeric output and monadic v
10
116
Upw
Printing width: affects all but bare output and error reports
(system dependent)
URL
Random link: used in ?
7*5
i. 1 + 2*31
OAI
Account information: identification, computer time, connect
time, keying time (all times in milliseconds and cumulative
during session)
UAV
Atomic vector
ULC
Line counter: statement numbers of functions in execution or
halted, most recently activated first
\0
UTS
Time stamp: year, month, day (of month) , hour (on 24-hour
clock), minute, second, millisecond
DTT
Terminal type: 1 for Selectric, 2 for PTTC/BCD, 3 for 1050
UUL
User Load
UWA
Working area available (in bytes)
Figure 4.2: System Variables
suspended.
[13
[2]
[3]
For example :
QLX+ ' F •
V F;ULX
ULX*-'+ULC,pU<r"WE CONTINUE FROM WHERE WE LEFT OFF'"
'WE NOW BEGIN LESSON 2'
DRILLFUNCTION V
)SAVE ABC
On the first activation of workspace ABC, the function F would be
automatically invoked; if it were later saved with F halted, subsequent
activation of the workspace would automatically continue execution from
the point of interruption.
The atomic vector UAV is a 256-element character vector,
containing all possible characters. Certain elements of UAV may be
terminal control characters, such as carrier return or linefeed, but
other elements of UAV may neither print nor exercise control. The
indices of any known characters can be determined by an expression such
as UAVx >ABCABC X .
Section 4: System Functions and System Variables 51
SECTION 5: SHARED VARIABLES
Two otherwise independent concurrently operating processors can
communicate , and thereby be made to cooperate, if they share one or more
variables. Such shared variables constitute an interface between the
processors, through which information may be passed to be used by each
for its own purposes. In particular, variables may be shared between
two active APL workspaces, or between an APL workspace and some other
processor that is part of the overall APL system, to achieve a variety
of effects including the control and utilization of devices such as
printers, card readers, magnetic tape units, and magnetic disk storage
units.
In use in an APL workspace, a shared variable may be either global
or local, and is syntactically indistinguishable from ordinary
variables. It may appear to the left of an assignment, in which case
its value is said to be set , or elsewhere in a statement, where its
value is said to be used . Either form of reference is an access .
At any instant a shared variable has only one value, that last
assigned to it by one of its owners. Characteristically, however, a
processor using a shared variable will find its value different from
what it might have set earlier.
A given processor can simultaneously share variables with any
number of other processors. However, each sharing is bilateral; that
is, each shared variable has only two owners. This restriction does not
represent a loss of generality in the systems that can be constructed,
and commonly useful arrangements are easily designed. For example, a
shared file can be made directly accessible to a single control
processor which communicates bilaterally with (or is integral with) the
file processor itself. In turn, the central processor shares variables
bilaterally with each of the using processors, controlling their
individual access to the data, as required.
It was noted in Section 4 that system variables are instances of
shared variables in which the sharing is automatic. It was not pointed
out, however, that access sequence disciplines are also imposed on
certain of these variables, although one effect of this was noted;
namely, variables like the time stamp accept any value specified, but
continue to provide the proper information when used. The discipline
that accomplishes this effect is an inhibition against two successive
accesses to the variable unless the sharing processor (the system) has
set it in the interim.
When ordinary, "undistinguished", variables are to be shared,
explicit actions are necessary to effect the sharing and establish a
desired access discipline. six system functions are provided for these
purposes: three for the actual management and three to provide related
information. These are summarized in Figure 5.1.
OFFERS
A single offer to share is of the form p USVO N, where p is the
identification of another processor and N is a character vector
representing a pair of names. The first of this pair is the name of the
variable to be shared, and the second is a surrogate name which is
offered to match a name offered by the other processor. The name of the
variable may be its own surrogate, in which case only the one name need
be used, rather than two. For example, the three sets of actions shown
below all have the same effect, which is to share one variable between
two processors 12 34 and 567 8, the variable being known to the former as
52 APL Language
FUNCTION
REQUIREMENTS [1]
EFFECT ON ENVIRONMENT
EXPLICIT RESULT
RANK
LENGTH
DOMAIN
P'QSVO N
22ppff
(x/pP)el ,~l+pff
Pe~l+i2*31
[2]
Tenders offer to processor
P if first (or only) name
of a pair is not previously
offered and not already in
use as the name of an ob-
ject other than a variable.
The offer is general (to
anyone) if 0=P.
Degree of coupling now in
effect for the name pair.
Dimension: x/ 1+pff.
DSVO N
2!>ppff
None.
[2]
None.
Degree of coupling now in
effect for the name pair.
Dimension: x/~i+pff.
C OSVC N
2>ppff
2>ppC
(l>ppC) Al=x/pC
or
(pC) = ( l + pJV),4
A/CeO 1
[2]
Sets access control.
New setting of access con-
trol.
Dimension: ( l+pff),4.
QSVC N
2!>ppff
None.
[2]
Existing access control.
USVR N
2>ppff
None.
[2]
Retracts offer (ends shar-
ing) .
Degree of coupling before
this retraction.
Dimension: x/'UpN.
QSVQ P
l>ppP
1*P,P
Pe - l+i2*31
None.
If 0=pP: Vector of identi-
fications of processors
making offers to this
user.
If l=x/pP: Matrix of names
offered by processor P
but not yet shared.
NOTES: 1. If a requirement is not met the function is not executed and a corresponding error
report is printed.
2. Each row of N (or N itself if 22ppff) must represent a name or pair of names. If a
pair of names is used for an offer (dyadic DSVO), either the pair, or the first
name only, can be used for the other functions.
Figure 5.1; Functions for the Management of Sharing
ABC, and to the latter as Q.
User 12 34 User 5 67 8
5678 QSVO 'ABC Y«
1 1234 DSVO 'Q Y'
5678 DSVO 'ABC Q*
1234 DSVO •«'
5678 DSVO MAC 1
12 34 DSVO iQ ABC 1
The surrogate names have no effect other than to control the
matching, making it possible for one processor to operate with no direct
knowledge of, or concern with, the variable name used by the other. The
same surrogate can be used in a succession of offers to the same
processor, in which case they are matched in sequence by appropriate
counter-offers. The same surrogate may also be used for offers to any
number of other processors at the same time. However, since a variable
may be offered to (or shared with) only one other processor at a time,
each coincident use of a particular surrogate name must be associated
with a different variable name.
Section 5: Shared Variables
53
The explicit result of the expression P DSVO N is the degree of
coupling of the name or name pair in N: zero if no offer has been made,
one if an offer has been made but not matched, two if sharing is
completed. An offer to any processor (other than the offering processor
itself) increases the coupling of the name offered if the name has zero
coupling and is not the name of a label, function, or group. An offer
never decreases the coupling.
The monadic function DSVO does not affect the coupling of the name
represented by its argument, but does report the degree of coupling as
its explicit result. If the degree of coupling is one or two, a
repeated offer has no further implicit result and either monadic or
dyadic DSVO may be used for inquiry. Advantage is taken of this in the
following example of a defined function for offering a name (to be
entered on request) to a processor P for a period of T seconds:
Z+P OFFER T;I;Q
m«- ' NAME : '
■+( • <K.=Q+a)/z+-i*-o
Il:-v(2 = Zf-P DSVO Q)/L2
■+(TzI+I+l + OxDDL 1)/L1
'NO DEAL*
+
L2: 'ACCEPTED' V
If the arguments of DSVO fail to meet any of the basic
requirements listed in Figure 5.1, the appropriate error report is
evoked and the function is not executed. If a user attempts to share
more variables than the quota allotted to him by those responsible for
the general management of the system the error report will be INTERFACE
QUOTA EXHAUSTED, and if, for any reason, the shared variable facility
itself is not available the report will be NO SHARES. An offer to a
processor will be tendered, whether or not the processor happens to be
available.
The value of a shared variable when sharing is first completed is
determined thus: if both owners had assigned values beforehand, the
value is that assigned by the first to have offered; if only one owner
had, that value obtains; if neither had, the variable has no value.
Names used in sharing are subject to the usual rules of localization.
A set of offers can be made by using a vector left argument (or a
scalar or one-element vector which is automatically extended) and a
matrix right argument, each of whose rows represents a name or name
pair. The offers are then treated in sequence and the explicit result
is the vector of the resulting degrees of coupling. If the quota of
shared variables is exhausted in the course of such a multiple offer,
none of the offers will be tendered.
An offer made with zero as left argument is a general offer, that
is, an offer to any processor. A general offer will be matched only
with a counter-offer which is not general, that is, one that explicitly
identifies the processor making the general offer. The processor
identification associated with a user is the user's, sign-on number.
Auxiliary processors are usually identified by numbers between 1 and
1000.
54 APL Language
ACCESS CONTROL
Consider the following simple example of sharing the variable 7
between two users 1234- and 567 8:
User 1234
User 5 67 8
5678 QSVO '7*
V+-5
V
123"+ QSVO '7'
7-*-3x7*2
75
The relative sequence of events in the two workspaces, after sharing, is
significant; for example, had the use of 7 by 1234 in the foregoing
example preceded the setting by 567 8, the resulting value would have
been 5 rather than 7 5 .
In most practical applications it is important to know that a new
value has been assigned between successive uses of a shared variable, or
that use has been made of an assigned value before a new one 'is set.
Since, as a practical matter, this cannot be left to chance, an access
control mechanism is embodied in the shared variable facility.
The access control operates by inhibiting the setting or use of a
shared variable by one owner or the other, depending upon the access
state of the variable, and the value of an access control matrix which
is set jointly by the two owners, using the dyadic form of the system
function QSVC . If, in the example above, one user (say 567 8) had
followed his offer to share 7 by the expression l l l 1 QSVC '7', then
the desired sequence would have been enforced. That is, the use of 7 by
5678 would be automatically delayed until 7 is set by 1234, and the use
by 1234 would be delayed until 7 is set by 5678.
The delay occasioned by the inhibition of any access uses only a
negligible amount of computer time. Interruption by a strong interrupt
signal during the period of delay aborts the access and unlocks the
keyboard .
Figure 5.2 shows
variable, the possible
inhibitions imposed by
ACM is associated with
second with its use.
indicated by the zeros
the access state shown
Figure 5.2 to validate
the three access states possible for a shared
transitions between states, and the potential
the access control matrix, ACM. The first row of
setting of the variable by each owner, and the
The permissible operations for any state are
in ACMaasm, where ASM is the representation of
in the figure. This can be confirmed by using
each of the following statements :
If ACMLl -,l] = l r then two successive sets by A require an intervening
access (set or use) by B.
If ACMll;2l=l, then two successive sets by B require an intervening
access by A.
If ACMi2;ll=li then two successive uses by A require an intervening
set by B.
If ACML2;2l=l, then two successive uses by B require an intervening
set by A.
The value of the access state representation is not directly
available to a user, but the value of the access control matrix is given
by the monadic function QSVC. For a shared variable 7 the result of the
Section 5: Shared Variables
55
Legend :
SA SB UA UB
Denote set or use by A
ACM: Access Control Matrix
ASM: Access State Matrix
or B.
A one in an element of ACM inhibits the
associated access. Allowable accesses are
given by the zeroes in ACM^ASM. Access control
vectors as seen by A and B, respectively, are
,ACM and AACM.
The access state matrix represents the last
access : ones occur in the last row if it is not
a set, and in a column if it is, the first
column if set by A and the last if set by B.
Figure 5.2: Access Control of a Shared Variable
56 APL Language
Access Control Vector
as seen by
Comments
A
B
No constraints.
Half -duplex. Ensures that each use is
preceded by a set by partner.
Half-duplex. Ensures that each set is
preceded by an access by partner.
Reversing half-duplex. Maximum
constraint.
Simplex. Controlled communication from
B to A. (For card reader, etc.)
11
110
1111
110
11
110
1111
10 1
Figure 5.3: Some Useful Settings for the Access Control Vector
expression USVC '7' executed by user A is the access control vector ,ACM
(the four-element ravel of ACM). However, if user B executed the same
expression he would obtain the result ,<$>ACM. The reason for the
reversal is that sharing is symmetric: neither owner has precedence
over the other, and each sees a control vector in which the first one of
each pair of control settings applies to his own accesses. This
symmetry is evident in Figure 5.2; if it were redrawn to interchange the
roles of A and B the control matrix would be the row-reversal of the
matrix shown.
The setting of the access control matrix for a shared variable is
determined in a manner which maintains the functional symmetry. An
expression of the form L USVC 'V executed by user A assigns the value
of the logical left argument L to a four-element vector which, for the
purposes of the present discussion, will be called QA. Similar action
by user B sets QB . The value of the access control matrix is determined
as follows:
ACM*-(2 2pQA)v§2 2pQB
Since ones in ACM inhibit the corresponding actions, it is clear from
this expression that one -user can only increase the degree of control
imposed by the other (although he can, by using USVC with a left
argument of zeros, restore the control to that minimum level at any
time) .
Access control can be imposed only after a variable is offered,
either before or after the degree of coupling reaches two. The initial
values of QA and QB when sharing is first offered are zero.
The access state when a variable is first offered (degree of
coupling is one) is always the initial state shown in Figure 5.2. If
the variable is set or used before the offer is accepted, the state
changes accordingly. Completion of sharing does not change the access
state.
Figure 5.3 lists a number of settings of the access control vector
which are of common practical interest. Any one of them can be
represented by a simplification of Figure 5.2 obtained by omitting the
control matrix and deleting the lines representing those accesses which
are inhibited in the particular case. For example, with maximum
constraints all the inner paths would be removed from the figure.
Section 5 : Shared Variables
57
A group of N access control matrices can be set at once by
applying the function OSVC to an if by t matrix left argument and an
tf-rowed matrix right argument of names. The explicit result is an N by
u- matrix giving the current values of the (ravels of) control matrices.
When control is being set for a single variable the left argument may be
a single 1 or. if all inhibits or none are intended.
RETRACTION
Sharing offers can be retracted by the monadic function DSVR
applied to a name or a matrix of names. The explicit result is the
degree (or degrees) of coupling prior to the retraction. The implicit
result is to reduce the degree of coupling to zero.
Retraction of sharing is automatic if the connection to the
computer is interrupted or if the user signs off or loads a new
workspace. Sharing of a variable is also retracted by its erasure or,
if it is a local variable, upon completion of the function in which it
appeared.
The nature of the shared-variable implementation is often such
that the current value of a variable set by a partner will not be
represented within a user's workspace until actually required to be
there. This requirement obtains when the variable is to be "used, when
sharing is terminated, or when a SAVE command is issued (since the
current value of the variable must be stored) . Under any of these
conditions it is possible for a WS FULL error to be reported. In all
cases the prior access state remains in effect and the operation can be
retried after corrective action.
INQUIRIES
There are three monadic inquiry functions which produce
information concerning the shared variable environment but do not alter
it; the functions QSVOand USVC already discussed, and the function OSVQ.
A user who applies the latter function to an empty vector obtains a
vector result containing the identification of each user making any
sharing offer to him. A user who applies the function OSVQ to a
non-empty argument obtains a matrix of the names offered to him by the
processor identified in the argument. This matrix includes only those
names which have not been accepted by counter-offers.
The expression (.Q*OSVO AO/Cl] M+ONL 2 can be used to produce a
character matrix whose rows represent the names of all shared variables
in the dynamic environment.
58 APL Language
SECTION 6: FUNCTION DEFINITION
There are three ways in which a defined function can be
established in an APL workspace:
1. It can be copied from a stored workspace using a system command,
as described in Section 8.
2. It can be established in execution mode, using the system
function DFX, either in direct keyboard entry or in the course
of execution of another defined function.
3. It can be established in function definition mode.
Regardless of which facility has been used for establishing a function,
its definition can be displayed or modified in either the function
definition mode, in which certain editing capabilities are built-in, or
by the combined use of the system functions OCR and QFX.
CANONICAL REPRESENTATION AND FUNCTION ESTABLISHMENT
The character representation of a function is a character matrix
satisfying certain constraints: the first row of the matrix represents
the function header and must be one of the forms specified below in the
section on function headers. The remaining rows of the matrix, if any,
constitute the function body , and may comprise any sequence of
characters. If the character representation satisfies additional
constraints such as no redundant spaces and left justification of the
non-blank character in each row, then it is said to be a canonical
representation .
Applying OCR to the character array representing the name of an
already established function will produce its canonical representation.
For example, if OVERTIME is an available function:
DEF+OCR 'OVERTIME*
DEF
PAI+R OVERTIME H;TIME
TIME+0\H-H0
PAY+R*1.5xTIME
pDEF
3 21
The function QCR applied to any argument which does not represent the
name of an unlocked defined function yields a matrix of shape 0.
Error reports for QCR are RANK error if the argument is not a vector or
a scalar, or DOMAIN error if the argument is not a character array.
The use of OCR does not change the status of the function
OVERTIME, which remains established and can be used for calculations.
Thus:
7 5 8 OVERTIME 3 5 40 4-5
60
If OVERTIME should be expunged:
DEX 'OVERTIME*
Section 6 : Function Definition 59
it is no longer available for use :
7 5 8 OVERTIME 3 5 40 45
SYNTAX ERROR
7 5 8 OVERTIME 3 5 40 4 5
A
The function can be re-established by UFXi
QFX DEF
OVERTIME
The function QFX produces as its explicit result the vector of
characters which represents the name of the function being fixed, while
replacing any existing definition of a function with the same name. The
function OVERTIME can now be used again:
7 5 8 OVERTIME 3 5 40 4 5
60
As noted in Section 4, an expression of the form UFX M will
establish a function if the following conditions are met:
1. M is a valid representation of a function. Any matrix which
differs from a canonical matrix only in the addition of
non-significant spaces (other than rows consisting of spaces only)
is a valid representation.
2. The name of the function to be established does not conflict with
an existing use of the name for an executing or halted function or
for a label, group, or variable.
If the expression fails to establish a function then no change occurs in
the workspace and the expression returns a scalar index of the row in
the matrix argument where the fault was found. If the argument of DFX
is not a matrix a RANK error will be reported, and if it is not a
character array a DOMAIN error will result.
THE FUNCTION HEADER
The v alence of a function is defined as the number of explicit
arguments which it takes. A defined function may have a valence of
zero, one, or two, and may yield an explicit result or not. These cases
are represented by six forms of header as follows:
Type Valence Result No Result
Dyadic 2 R+-A F B A F B
Monadic i r+f B F B
Niladic R+F F
The names used for the arguments of a function become local to the
function, and additional local names may be designated by listing them
after the function name and argument, separated from them and from each
other by semicolons; the name of the function is global . The
significance of these distinctions is explained below.
Except that the function name itself may be repeated in the list
of local names, a name may not be repeated in the header. It is not
obligatory either for the arguments of a defined function to be used
within the body, or for the result variable to be specified in the
course of function execution.
60 APL Language
LOCAL AND GLOBAL NAMES
In the execution of a defined function it is often necessary to
work with intermediate results or temporary functions which have no
significance either before or after the function is used. The use of
local names for these purposes, so designated by their appearance in the
function header, avoids cluttering the workspace with a multitude of
objects introduced for such transient purposes, and allows greater
freedom in the choice of names. Names used in the function body, and
not so designated, are said to be global to that function.
A local name may be the same as that for a global object, and any
number of names local to different functions may be the same. During
the execution of a defined function, a local name will temporarily
exclude from use a global object of the same name. If the execution of
a function is interrupted (leaving it either suspended , or pendent (See
Section 7) , the local objects retain their dominant position, during the
execution of subsequent APL operations, until such time as the halted
function is completed. However, system commands and the del form of
function definition (see below) continue to reference global objects
under these circumstances.
The localization of names is dynamic, in the sense that it has no
effect except when the defined function is being executed. Furthermore,
when a defined function uses another defined function during its
execution, a name localized in the first (or outer) function continues
to exclude global objects of the same name from the purview of the
second (or inner) function. This means that a name localized in an
outer function has the significance assigned to it in that function when
used without further localization in an inner function. The same name
localized in a sequence of nested functions has the significance
assigned to it at the innermost level of execution.
The shadowing of a name by localization is complete, in the sense
that once a name has been localized its global and outer values are
nullified, even if no significance is assigned to it during execution of
the function in which it is localized.
BRANCHING AND STATEMENT NUMBERS
Statements in a function are normally executed successively, from
top to bottom, and execution terminates at the end of the last statement
in the sequence. This normal order can be modified by branches .
Branches are used in the construction of iterative procedures, in
choosing one out of a number of possible continuations, or in other
situations where decisions are made during the course of function
execution.
To facilitate branching, the successive statements in a function
definition have reference numbers associated with them, starting with
the number one for the first statement in the function body and
continuing with successive integers, as required. Thus, the expression
+4 denotes a branch to the fourth statement in the function body, and
when executed causes statement t to be executed next, regardless of
where the branch statement itself occurs. (In particular -+-M- may be
statement 4, in which case the system will simply execute this "tight
loop" indefinitely, until interrupted by an action from the keyboard.
This is a trap to be avoided.)
A branch statement always starts with the branch arrow (or right
arrow ) on the left, and this can be followed by any expression. For the
statement to be effective, however, the expression must evaluate to an
integer, to a vector whose first element is an integer, or to an empty
vector; any other value results in a DOMAIN or RANK error. If the
Section 6: Function Definition 61
expression evaluates to a valid result, then the following rules apply:
1 . If the result is an empty vector the branch is vacuous and
execution continues with the next statement in the function if
there is one, or else the function terminates.
2. If the result is the number of a statement in the function then
that statement is the next to be executed.
3. If the result is a number out of the range of statement numbers
in the function, then the function terminates. The number o and
all negative integers are outside of the range of statement
numbers for any function.
Since zero is often a convenient result to compute, and it is not
the number of a statement in the body of any function, it is often used
as a standard value for a branch intended to end the execution of a
function. It should be noted that in the function definition mode
described below, zero is used to refer to the header. This has no
bearing on its use as a target for a branch.
An example of the use of a branch statement is shown in the
following function, which computes the greatest common divisor of two
scalars:
Z+M GCD N
Z+M
M+M\N
N+Z
-*(0*M)/1
The compression function in the form U/V gives V if U is equal to
one, and an empty vector if U is equal to zero. Thus, the fourth
statement in GCD is a branch statement which causes a branch to the
first statement when the condition q*m is true, and a branch with an
empty vector argument, that is, normal sequence, when the condition is
false. In this case, there is no next statement and so execution of the
function ends.
LABELS
If a statement occurring in the body of a function definition is
prefaced by a name and a colon, then the name is assigned a value equal
to the statement number. A name used in this way is called a label .
Labels are used to advantage when it is expected that a function
definition may be changed for one reason or another, since a label
automatically assumes the new value of the statement number of its
associated statement as other statements are inserted or deleted.
The name of a label is local to the function in which it appears,
and must be distinct from other label names and from the local names in
the header.
A label name may not appear immediately to the left of a
specification arrow. In effect it acts like a (local) constant.
COMMENTS
The lamp symbol pi (the cap-null) signifies that what follows it is
a comment, for illumination only and not to be executed; it may occur
only as the first character in a statement.
62 APL Language
FUNCTION EDITING - THE V FORM
The functions QCR and QFX together form a basis for establishing
and revising functions. Convenient definition editing with them,
however, requires the use of prepared editing functions which must be
defined, stored in a library, and explicitly activated when needed. The
del form described here provides an alternative facility for function
entry and revision which is always present for use.
When the user enters the del character (v) followed by the name of
a defined function, the system responds by displaying [tf+1] , where N is
the number of statements in the function. It is now possible to:
add, insert, or replace statements
replace the header
modify the header or a statement
delete a statement
display all or part of the definition
A new function is started by entering the desired header on the
same line as the opening V. Once the function definition mode has thus
been entered, the treatment of a new function is identical to that for a
function already defined.
ADDING A STATEMENT
If the response to the display of statement number [tf+1] is a
statement, it is accepted as a line added at the end of the definition.
The system response is the display [W+2] . Additional statements may
continue to be added to the definition in this manner. If an empty
statement is entered, the system will re-display the line number in
brackets.
INSERTING OR REPLACING A STATEMENT
If the response to the statement number displayed by the system is
IN] , where N is any positive number with or without a fractional part,
the system will display INI . A statement entered will replace an
existing statement N.
The system continues by displaying the next appropriate number.
For example, if the statement number entered was [3] , the next number
displayed will be [4] ; if [3.02] , then [3.03] ; if [3.29] then [3.3] ,
and so forth.
A statement may be submitted with line number [#]; it will be
inserted or will replace an existing statement in the manner described.
The response of the system in this case is to display the next statement
number.
REPLACING THE HEADER
If the user enters [0] , the system responds by displaying [0] .
The user may now enter any legal header, which will replace the existing
header. Following this, the system displays [1] . The entire operation
may be carried out by entering [0] and, on the same line, the header.
DELETING A STATEMENT
One or more statements may be deleted by entering in brackets a
delta followed by one or more statement numbers, for example, [A2 7 3].
The response of the system is to display the statement number that
follows the last number in the list. In the example, the response will
be [4].
Section 6: Function Definition 63
EDITING A STATEMENT OR HEADER
Statement N can be modified by the following mechanism:
1. Enter [AW] where M is an integer. The header is referred to
by using o for N.
2. Statement N or the header is automatically displayed and the
cursor stops under position M, which is counted from the left
margin.
3. A letter or decimal digit or the symbol / may be typed under
any of the positions in the display. Any other characters
typed in this mode are ignored. The ordinary rules for
editing before entry (such as backspace and linefeed) apply.
4. The line is re-displayed. Each character understruck by a /
is deleted, each character understruck by a digit K is
preceded by K added spaces, and each character understruck by
a letter is preceded by 5*R spaces, where R is the position of
the letter in the alphabet. For example, the letter C will
produce 15 spaces. Finally, the cursor moves to the first
injected space and awaits the typing of modifications in the
usual manner . The final effect is as if the entry had been -
made entirely from the keyboard; in particular, a completely
blank sequence leaves the line unchanged.
If a statement number itself is changed from N to P during the
editing procedure the statement affected is determined by the new
statement number P; hence statement N remains unchanged. This permits
statements to be moved, with or without modification.
If an attempt to display a line would exceed the printing width
(determined by DPW) this form of editing is not applicable. Such a
condition may arise if the line is too long to start with, or if too
many spaces have been requested for insertion.
ADDING TO A STATEMENT OR HEADER
One or more characters can be added to the end of statement N, or
statement N can be corrected, by entering [iVDO]. In response, the
system displays statement N; the cursor is positioned at the end of the
statement, and the keyboard unlocks. The statement may be extended, or
modified by using the normal revision procedures for entry. In
response, the system displays the next statement number and awaits
entry.
The header may be modified in this way by entering [0[]0].
FUNCTION DISPLAY
The canonical representation of a function includes the header and
body displayed as a character matrix. The V form permits display of a
canonical representation modified as follows:
1. Labelled lines and comments are offset one space to the left.
2. Statement numbers in brackets are appended to the left of the
statements.
3. A del character (7) is prefixed to the header, separated by
one space.
4. A final line is added, consisting of spaces and a del
character, aligned with the del character which prefixes the
header.
64 APL Language
Shown are the canonical representation and function display of a
function for computing the determinant of a matrix:
UCR'DET' VDETLOlV
Z+DET A;B;P;I V Z+DET A;B;P;I
I+QIO [ 1 3 I+QIO
Z«-l [2] Z<-1
L:F«-( U[;J])ir/UC;I] [3] L : P-K \A C ;I ] ) i \ I \A C ;I]
+(P=I)/LL [4] -*{P=I)/LL
AiI,P;l<-AlP,I;l [5] 4[I ,P; l-MLP.I ; ]
Z«--Z [6] Z-e--Z
LL:Z-e-ZxBfi4[I;J] [7] LL : Z+-ZXB+-4 [ J ; J]
+ (0 1 v.=Z,l + p4)/0 [8] +(0 1 v.=Z,l + pi4)/0
4 + 1 1 +4-U[;J3*B)°.x4[I; ] [9] 4+1 1 + A- {A [ ;X]tB ) <> . x4 [I ; ]
-»-£ [10] ->L
^EVALUATES A DETERMINANT [11] ^EVALUATES A DETERMINANT
V
The display of a function can be obtained by typing CD]. After
the display, the system displays [ff+1] where ff is the number of
statements in the function.
Statement N can be displayed by typing [ffQ]. After display the
system types INI. Entering LQKl causes display of all statements from
K onwards. After the display the system displays IN +11 where N is the
number of statements in the function.
LEAVING TIfE V FORM
The del form may be left by typing a V on a line by itself, or
as the last character on any entry except one which enters a comment
statement.
In particular, it can follow a request for display or a function
statement, and either can be incorporated in the same entry that both
opens and closes the definition mode, as in VDETlQlV or VDETilOl+L ,LV.
On leaving the del form, the statements are reordered according to
their statement numbers and the statement numbers are replaced by the
integers 1, 2, 3, and so on.
A function definition can be locked by either opening or closing
the definition mode with a del-tilde, V. The use of this is explained
in Section 7.
Section 6: Function Definition 65
SECTION 7: FUNCTION EXECUTION
A defined function may be used like a primitive function, except
that it may not be the argument of a primitive operator. In particular,
a defined function may be used within its own definition or that of
another defined function. When a function is called, or put into use,
its execution begins with the first statement, and continues with
successive statements except as this sequence is altered by branch
instructions .
Consider the function OVERTIME:
PAY+R OVERTIME H-.TIME
TIME^0[^0-H
PAY*- Rxl.5*TIME
If this function is invoked by a statement such as X OVERTIME I the
effect is to assign to the local name R the value of X and to H the
value of Y, and then execute the body of the function OVERTIME. Except
for having a value assigned initially, the argument variable is treated
as any other local variable .and, in particular, may be respecified
within the function.
A function like OVERTIME, which produces an explicit result, may
properly be used in compound expressions. In the OVERTIME function the
last value received by PAY in the course of execution is the explicit
result of the function. For example:
YTDAT
100 200 150
YTDAT+YTDAT+0T+-5 7 6 OVERTIME 3 5 40 4 5
OT
45
YTDAT
100 200 195
PAY, itself, is a local variable and therefore has no significance after
the function is executed:
PAY
VALUE ERROR
PAY
HALTED EXECUTION
The execution of a function F may be stopped before completion in
a variety of ways: by an error report, by an attention signal, or by the
stop control treated below. When this happens the function is said to
be suspended , and its progress can be resumed by entering a branch-
statement from the keyboard. Whatever the reason for suspension, the
name of the function is displayed, with a statement number beside it.
In the case of an error stop or an interrupt the statement itself is
also displayed, with an appropriate message and an indication of the
point of interruption. Unless a specification appears in the statement
to the right of this point, the state of the computation has been
restored to the condition obtaining before the statement started to
execute .
In general, therefore, the displayed number is that of the
statement that should be executed next if the function is to continue
normally. Resumption of execution at that point can be accomplished by
entering a branch to that number specifically, a branch to an empty
vector, or a branch to QLC. Entering -*0 , or a branch to another number
66 APL Language
outside the range of statement numbers, causes an immediate exit from
the function and it is no longer suspended.
In the suspended state all normal activities are possible, but
names used refer to their local significance, if any. The system is in
a condition to execute statements or system commands, resume execution
of the function at an arbitrary point, or enter definition mode to work
on any function which is not pendent (see below) . In particular, the
suspended function can be edited at this point (if it is not itself
pendent from a previous use) .
STATE INDICATOR
Entering the system command )SI causes a display of the state
indicator ; a typical display has the following form:
)SI
Hill *
F[2]
G[3]
This display indicates that execution was halted before completing
(perhaps before starting) execution of statement 7 of function H, that
the current use of function H was invoked in statement 2 of function F,
and that the use of function F was in turn invoked in statement 3 of C.
The * appearing to the right of ff[7] indicates that the function H is
suspended. The functions G and F are said to be pendent , because their
execution cannot be restarted directly, but only as a consequence of
function H resuming its course of execution. The term halted is used
to describe a function which is either pendent or suspended.
Further functions can be invoked when in the suspended state.
Thus, if G were now invoked and a further suspension occurred in
statement 5 of Q , itself invoked in statement 8 of G, a subsequent
display of the state indicator would appear as follows:
)SI
«L5]
*
GC8]
ff[7]
*
GC2]
F[3]
Since the line counter, ULC , holds the current statement numbers
of functions that are in process of execution, its value at this point
would be the vector 5 8 7 2 3. The sequence from the last to the
preceding suspension can be cleared by entering a right arrow (->■) . This
behavior is illustrated by continuing the foregoing example as follows!
fl[7]
G[2]
**[3]
7 2 3
)SI
ULC
Repeated use of •*■ will clear the state indicator completely, and restore
ULC to an empty vector. The cleared state indicator displays as a blank
line.
Section 7: Function Execution 67
STATE INDICATOR DAMAGE
If the name of a function occurs in the state indicator list, then
erasure of the function or replacement of the function by copying a
function with the same name (even another instance of the same function)
will make it impossible for the original course of execution to be
resumed. In such an event an SI DAMAGE report is given. In addition,
some APL systems will give an SI DAMAGE report if a suspended function
is edited to change the order of its labels or to modify its header.
If an SI DAMAGE report is given for a suspended function, it will
not be possible to resume its execution by entering a branch statement,
but the function can be invoked anew, with or without prior clearance of
the state indicator.
In case of SI DAMAGE, display of the state indicator will show the
damage, possibly by a blank where the function name should have appeared
or by some explicit indicator or both.
TRACE CONTROL
A trace is an automatic display of information generated by the
execution of a function as it progresses. In a complete trace of a
function, the number of each statement executed is displayed in
brackets, preceded by the function name and followed by the final value
produced by the statement. The trace of a branch statement shows a
branch arrow followed by the number of the next statement to be
executed. The trace is useful in analyzing the behavior of a defined
function, particularly during its design.
The tracing of a function PROFIT. is controlled by the trace
control for PROFIT, denoted by TLPROFIT. If one sets T A PR FIT*- 2 3 5
then statements 2, 3, and 5 will be traced in any subsequent execution
of PROFIT. TLPROFIT+xQ discontinues tracing of PROFIT. A complete
trace of PROFIT is obtained by ThPROFIT+xN , where N is the number of
statements in PROFIT. In general, the trace control for any function is
designated by prefixing TA to the function name.
STOP CONTROL
A function can be caused to execute up to a certain statement and
then stop in the suspended state. This is frequently useful in
analyzing a function, for example by experimenting with local variables
or intermediate results. The stops are set by the stop control in the
same manner as the trace. For a function PROFIT, stops before "lines 4
and 12 are executed would be set by entering S&PROFIT+H 12.
At each stop, the function name and line number are displayed as
described above for suspended functions. To go to the next stopping
point after the first, execution must be explicitly restarted by
entering an appropriate branch statement.
Trace control and stop control can be used in conjunction.
Moreover, either of the controls may be set within functions. In
particular, they may be set by expressions which initiate tracing or
stops as a result of certain conditions that may develop in the course
of function execution, such as a particular variable taking on a
particular value. They may only be used as the left argument of
specification. They may not be used by themselves or as the argument to
a function .
68 APL Language
LOCKED FUNCTIONS
If the symbol ¥ (called del-tilde) is used instead of v to open or
close a function definition, the function becomes locked . A locked
function cannot be revised or displayed in any way. Any associated stop
control or trace control is nullified after the function is locked.
A locked function is treated essentially as a primitive, and its
inner workings are concealed as much as possible. Execution of a locked
function is terminated by any error ocurring within it, or by a strong
interrupt. If execution stops the function is never suspended but is
immediately abandoned. The message displayed for a stop is DOMAIN error
if an error of any kind occured, WS FULL and the like if the stop
resulted from a system limitation, or INTERRUPT.
Moreover, a locked function is never pendent, and if an error
occurs in any function invoked either directly or indirectly by a locked
function, the execution of the entire sequence of nested functions is
abandoned. If the outermost locked function was invoked by an unlocked
function, that function will be suspended; if it was invoked by a
keyboard entry the error message will be displayed with a copy of that
statement.
Similarly, when a weak interrupt is encountered in a locked
function, or in any function that was ultimately invoked by a locked
function, execution continues normally up to the first interruptable
point: either the next statement in an unlocked function that invoked
the outermost locked function, or the completion of the keyboard entry
that used this locked function. In the latter case, the weak interrupt
has no net effect.
Locked functions may be used to keep a function definition
proprietary, or as part of a security scheme for protecting other
proprietary information. They are also used to force the kind of
behavior just described, which sometimes simplifies the use of
applications .
RECURSIVE FUNCTIONS
A defined function whose name has not been made local and is used
in the body of the function definition is said to be defined
recursively . For example, one definition of the greatest common divisor
function states that the greatest common divisor of zero and any number
N is N; for any other pair of numbers it is the greatest common divisor
of the residue of the second number by the first, and the first number.
The words "greatest common divisor" are used in the definition. This
suggests that a greatest common divisor function GCDR can be written
whose canonical representation is:
DCR * GCDR '
R+A GCDR B
R+B
-Ko=4)/o
R+(A |B) GCDR A
18 GCDR 45
9
This can be compared to the equivalent function defined iteratively in
Section 6.
Executing an erroneously defined recursive function will often
result in a WS FULL report. The non-trivial execution of a properly
defined recursive function may also have this effect because of the very
deep nesting of function calls that is often required.
Section 7: Function Execution 69
TERMINAL INPUT AND OUTPUT
In many significant applications, such as text processing, for
example, it is necessary that the user supply information as the
execution of the application programs progresses. It is also often
convenient, even in the use of an isolated function, to supply
information in response to a request, rather then as arguments to the
function as part of the original entry. This is illustrated by
considering the use of the function CI, which determines the growth of a
unit amount invested at periodic interest rate R for number of periods
T:
OCR' CI 1
A+R CI T
A<-(1+R)*T
For example, the value of 1000 dollars at 5 per cent for 7 years,
compounded quarterly, might be found by:
1000 x (.05*4) CI 7x4
1415.992304
The casual user of such a function might, however, find it difficult to
remember which argument of CI is which, how to adjust the rate and
period stated in years for the frequency of compounding, and whether the
interest rate is to be entered as the actual rate (for example, .05) or
as a percentage (for example, 5). An exchange of the following form
might be more suitable:
INVEST
ENTER CAPITAL AMOUNT IN DOLLARS
D:
1000
ENTER NUMBER OF TIMES COMPOUNDED IN ONE YEAR
□ :
4
ENTER ANNUAL INTEREST RATE IN PERCENT
D:
5
ENTER PERIOD IN YEARS
D:
7
VALUE IS 1415.992304
It is necessary that each of the entries (1000, 4, 5, and 7)
occurring in such an exchange be accepted not as an ordinary entry
(which would only evoke the response 1000, etc.), but as data to be used
within the function INVEST. Facilities for this are provided in two
ways, termed evaluate d input , and character input . A definition of the
function INVEST, which uses evaluated input, is as follows:
UCR' INVEST'
INVEST ;CiR;T;F
' ENTER CAPITAL AMOUNT IN DOLLARS*
C*-D
'ENTER NUMBER OF TIMES COMPOUNDED IN ONE YEAR 1
F+0
'ENTER ANNUAL INTEREST RATE IN PER CENT'
R+U+FxlOO
'ENTER PERIOD IN YEARS'
T-f-FxQ
'VALUE IS ' ,vCxR CI T
The function NEWSTOCK in the Introduction (Section 1) is an example of
the use of both character input and bare output (discussed below) .
70 APL Language
EVALUATED INPUT
The quad symbol Q appearing anywhere other than immediately to the
left of a specification arrow denotes request for keyboard input as
follows: the two symbols D: are displayed, and the keyboard is unlocked
on the next line, indented from the left margin. Any valid expression
entered at this point is evaluated, and the result substituted for the
quad. Suppose F is a function whose definition includes a quad symbol:
OCR ' F '
Z-f-F
Z-s-l+xQ
F
D:
3 + 2
20
An invalid entry in response to a request for quad input causes an
appropriate error report, after which input is again awaited. For
example, entering an expression which has no result produces a value
error. Function definition mode (the editing or display of functions,
or creation of new functions) is not permitted during Q entry. In
general, a system command entered during Q input is executed, but the
system's response to the command is not treated as a response to D-
After execution of a command, valid input is again awaited (unless the
command was one which replaced the contents of the active workspace) .
An empty input (one containing nothing other than zero or more spaces)
is rejected and the system again awaits input.
INTERRUPTING EVALUATED INPUT
Execution of a defined function containing a request for Q input
can be interrupted at the statement containing the D by entering a
statement which has no value (for example <t ' ' or a branch statement, or
any defined function which does not return a result) . This suspends
execution of the statement containing the U, with the report VALUE
ERROR. A command that requires saving the workspace (either SAVE or
CONTINUE, described in Section 8), issued during a request for □ input,
also interrupts the statement containing the Q before carrying out the
command .
The response ■+ entered in response to a Q abandons execution of
the function and any pendent functions leading up to it.
CHARACTER INPUT
The quote-quad symbol H (that is-, a quad overstruck with a quote)
is a request for character input: entry is permitted at the left margin
and data entered are accepted as characters. For example:
X-s-B
CAN'T (Quote-quad input, not indented)
X
CAN'T
INTERRUPTING CHARACTER INPUT
Request for B input can be interrupted,, in the case of certain
typewriter terminals, by entering the three letters OUT, in that order,
but with a backspace between each pair so that they all overstrike.
This interrupts execution, but does not cause an exit from a defined
function.
Section 7: Function Execution 71
NORMAL OUTPUT
The quad symbol appearing immediately to the left of a
specification arrow indicates that the value of the expression to the
right of the arrow is to be displayed in the standard format (subject to
the printing precision DPP and the printing width QPW) . Hence, Q+x is
equivalent to the statement X. The longer form Q+x is useful when
employing multiple specification. For example, U+Q+X*2 assigns to Q the
value X*2 and then prints the value of J* 2.
The maximum length of a line of normal display (measured in
characters) is called the printing width , and is given by the value of
the system variable QPW. The maximum and minimum effective values of
the printing width depend upon the system being used, and the
appropriate user's guide should be consulted. A display whose lines
exceed the printing width is ended at or before the maximum length, and
continued on subsequent lines.
BARE OUTPUT
Normal output includes a concluding new line signal so that the
succeeding display (either input or output) will begin at a standard
position on the following line. Bare output, denoted by expressions of
the form Ch-X does not include this signal if it is followed either by
another bare output or by character input (of the form X+B) .
Character input following a bare output is treated as though the
user had spaced over to the position occupied at the conclusion of the
bare output, so that the characters received in response will normally
be prefixed by a number of space characters. This allows for the
possibility that, after the keyboard is unlocked, the user backspaces
into the area occupied by the preceding output. The following function
prompts the user with whatever message is supplied as its argument, and
evaluates the response:
UCR' PROMPT*
Z+-PROMPT MSG
B+-MSG
Z-HD
Using such a function, the expression:
PROMPT 'ENTER CAPITAL: •
would have the following effect:
displayed by system
ENTER CAPITAL: "lOOO
entered by user
The value of Z is as many blank characters as there are characters
in MSG, followed by the characters entered by the user. In this case Z
will have fifteen blanks followed by the character '1000'.
The new line signals that would be supplied by the system in order
to break lines that exceed the printing width are not supplied with bare
output. However, since an expression of the form E+X entered directly
from the keyboard (rather than being executed as part of a defined
function) must necessarily be followed by another keyboard' entry, the
output it causes is concluded with a new line signal. The effect is in
this case indistinguishable from normal output, except for the
possibility of exceeding the printing width limitation.
72 APL Language
SECTION 8: SYSTEM COMMANDS
An APL system recognizes two broad classes of instructions,
statements and system commands . System commands control the initiation
and termination of a work session, saving and reactivating copies of a
workspace, and transferring data from one workspace to another.
System commands can be invoked only by individual entries from the
keyboard and cannot be executed dynamically as part of a defined
function. They are prefixed by a right parenthesis.
They will be
The system commands are summarized in Figure 8.1
discussed under three main headings:
1. The active workspace.
a. Action.
b. Inquiry.
2. Workspace storage and retrieval.
a. Action.
b. Inquiry.
3. Access to the system.
A command that is not recognizable, or is improperly formed, is
rejected with the report INCORRECT COMMAND. Certain commands may also
result in more specific trouble reports; these are discussed in the
appropriate context and are summarized in Figure 8.2.
Once its execution has started, a system command cannot be
interrupted, although display of the system's response to the command
can be suppressed by an interrupt signal.
In the text that follows, each system command is shown in a sample
form. In use, the appropriate names or numbers should, of course, be
substituted for those shown.
A
N
USERNO
LIBNO
WSNAME
GRPNAME
OB J NAME
PASS WD
NEWPASS
[ ]
A letter of the alphabet.
A number.
A user account number.
A library number (that is, either a
user account number or a public
library number) .
A workspace name.
A group name.
A name of an object within a
workspace (that is, a function, a
variable, or a group) .
A password, which must match the
previously set value.
A new password, which need not match
the previously set value.
Items enclosed in brackets may in
some circumstances be omitted.
Section 8: System Commands 73
FORM
PURPOSE
NORMAL RESPONSE
TROUBLE REPORTS
TERMINAL CONTROL COMMANDS
)number [pass]
Identify user and start use of APL
[public-adc
header
system
Iress]
1 5
12 13 14
[SAVED time date]
^CONTINUE [newpass]
Replace ws CONTINUE by copy of active ws and
end use of APL
[time date
header
account
CONTINUE]
4 1t
) CONTINUE HOLD [newpass]
Replace ws CONTINUE by copy of active ws, end
use of APL, but hold connection
[time date
header
account
CONTINUE]
4 M
)OFF [newpass]
End use of APL
header
account
header
4
)OFF HOLD [newpass]
End use of APL, but hold connection
4
account
ACTIVE WORKSPACE CONTROL COMMANDS
) CLEAR
Activate a clear ws
CLEAR WS
4
)COPY wsid [pass]
Copy all global objects from named ws into
active ws
SAVED time
date
2 3
4 16 17 18 19 20
)COPY wsid [pass] names
Copy global objects named from designated ws
into active ws
SAVED time
date
2 3
4 7 16 17 18 19 20
) ERASE [names].
Erase global objects named from active ws
4 7
16
) GROUP names
Gather objects into (or disperse) a group
(first name designates group)
4 8
17 18
)LOAD wsid [pass]
Activate copy of named ws
SAVED time
date
3 4
18 19
)PCOPY wsid [pass]
Copy all objects from designated ws not named
in active ws
SAVED time
date
2 3
4 6 16 17 18 19 20
)PCOPY wsid [pass] names
Copy objects designated that are not named in
active ws
SAVED time
date
2 3
4 « 7 16 17 18 19 20
) SYMBOLS number
Change maximum number of symbols in ws to
number
WAS number
4
)WSID wsid [newpass]
Change identification of active ws
WAS wsid
4
LIBRARY CONTROL COMMANDS
)DROP wsid
Drop ws from library
time date
3 4
20
)SAVE
Place copy of active ws in library
time date wsid
3 4
9 10 11 18
)SAVE wsid [newpass]
Replace named ws by copy of active ws
time date
3 4
9 10 11 18
INQUIRY COMMANDS
) FNS [alphabetic]
List defined functions (whose initials follow
given character in alphabet)
[names]
4
)GRP name
List members of named group
[names]
4 V
)GRPS [alphabetic]
List groups (whose initials follow
given character of alphabet)
[names]
4
)LIB [number] [alphabetic]
List workspaces in designated library (whose
Initials follow given character in alphabet)
[names]
3 4
)SI
List halted functions
state-indicator
4
)3IV
List halted functions and associated local
state-indicator and
4
names
names
) SYMBOLS
Give maximum number of names in ws
IS number
4
)VARS [alphabetic!
List global variables (whose initials follow
given character in alphabet)
[names]
4
)WSID
Give wsid of active ws
[number] name
4
Notes: 1. Items in brack*
sts are optional
2. Abbreviations <
snd meanings:
ws \
7orkspace
pass <
s colon possibly followed by a password
newpass <
i pass that does not have to match a previous p
as sword
header t
i port number, time of day, date, and user-code
account <
<
rives the connect time and compute time since 1
ind since beginning of the accounting period
ast start
wsid <
i ws name possibly preceded by a library number
public-address
a message to users of the system
3. The commands )l
IRASEt )FNS. and )VARS have variants that are s
ystem functions
(see Qgx and Qj
?C) .
Figure 8.1: System Commands
74 APL Language
No.
TROUBLE REPORT
1 ALREADY SIGNED ON
2 DEFN ERROR
3 IMPROPER LIBRARY REFERENCE
<* INCORRECT COMMAND
5 INCORRECT SIGN -ON
6 NOT COPIED: names
7 NOT FOUND : purported names
8 NOT GROUPED. NAME IN USE
9 NOT SAVED, THIS WS IS CLEAR
10 NOT SAVED, THIS US IS wsid
11 NOT SAVED, WS QUOTA USED UP
12 NUMBER IN USE
13 NUMBER LOCKED OUT
14 NUMBER NOT IN SYSTEM
15 RESEND
16 SI DAMAGE
17 SYMBOL TABLE FULL
18 WS FULL
19 WS LOCKED
20 WS NOT FOUND
Meaning
WS
APL already in use at terminal
Attempted copy or protected copy of
function definition as response to
□ input request (on some systems)
1) Number is not a library number, or
2) Attempted save into alien library,
or 3) Attempted reference to alien
CONTINUE WS
Global homonyms in active ws are
protected
Ws does not contain global objects
with purported names
First name is name of a global
function or variab.le
A clear ws has no name and cannot be
stored
Attempted replacement of a stored ws
whose identification does not match
that of the active ws
Allotted number of stored wss
previously reached
Authorization for use of number has
been withdrawn
1) Number entered is not an account
number, or 2) Password missing, or
3) Wrong password used
Transmission failure or more than the
implementation allowed number of
characters entered in one line
State indicator damaged while
performing an ) ERASE or )C0PY command
Too many names used
Remedy
1) To display account number, use OAI
2) To change account number, execute
)0FF HOLD or ) CONTINUE HOLD and
sign on
Workspace full, possibly because of
1 ) Temporary values produced during
evaluation of an expression, or
2) Value assigned to shared variable
by partner.
1) Password missing, or
2) Wrong password used
No stored ws with given identification
1) Use different name for group, or
2) Erase global object if not needed
Remove active ws, then store
1 ) Drop an unneeded ws , or
2) Ask APL operator to increase quota
Consult APL operator
Consult APL operator
Consult APL operator
If chronic, redial or have terminal
or phone repaired
Erase objects not needed, save ws,
clear active ws, and perhaps change
limit, using )SYMB0LS, copy the
saved ws and rename the active ws
1) Erase objects not needed, or
2) Clear state indicator, or
3) Revise method of calculation
Figure 8.2: Trouble Reports
Section 8:
System Commands 75
Items enclosed in angles may in some
circumstances be omitted, and, where
omitted, the system supplies values
from the existing user
identification, workspace
identification, or previously
established password.
COMMANDS THAT MODIFY THE WORKSPACE ENVIRONMENT
The following system commands affect the active workspace, the
environment in which computation takes place and in which names have
meaning. In particular, the active workspace contains the settings of
the state indicator (discussed in Section 7) and other elements of the
computing environment, mediated by several of the system variables
(discussed in Section 4) .
) CLEAR
This command is used to make a fresh start, discarding the
contents of the active workspace, and resetting the
environment to standard initial values (see Figure 8.3). At
sign-on, the user receives a clear workspace characterized by
these same, initial values, unless the workspace CONTINUE was
automatically loaded.
Symbol table size
Index origin, 010
Latent expression, QLX
Line counter, QLC
State indicator
Workspace name
Workspace password
Printing precision, DPP
Printing width, UPW
Comparison tolerance, OCT
Random link, ORL
Work area available, QWA
* 256-
1
Empty
Empty
Cleared
None (CLEAR WS)
None
* 10
* 120
* 1E~13
* 16807
* Depends on the local
installation, and, in
some systems , upon
options selected by
the user.
* Items marked with an asterisk have values which
may vary from system to system. The values shown
are widely used.
Figure 8.3. Environment Within a Clear Workspace
76 APL Language
) SYMBOLS N
Sets the size of the symbol table, that is, the maximum number
of names that may occur in the workspace. (Note that the
occurrence of a name includes not only the names of functions,
variables, or groups themselves, but also any names occurring
within their definitions) . New values of the maximum may be
set only in a clear workspace. An attempt to change the
maximum once the workspace is no longer clear, or to set it
outside the range permitted by the system, is rejected with
the report INCORRECT COMMAND. Valid use of the command
results in the report WAS . . . , showing the former limit.
)ERASE 10BJNAME1 0BJNAME2 0BJNAME3 ...]
The global objects named are expunged from the workspace;
shared variable offers with respect to any of them are
retracted. If a name is the name of a group (see below) , the
group and all of its members present in the workspace are
erased. (If a member of a group is itself a group, xts
definition is erased but not its members.)
If a halted function is erased, the report SI DAMAGE is
displayed, and the name of that function is replaced in the
state indicator by blanks. It is not possible to resume the
execution of an erased function, and the user should enter ■*■
one or more times to clear the state indicator of indications
of damage.
If an object named in the command cannot be found, the report
NOT FOUND: is emitted, followed by a list of the objects not
found .
)COPY <LIBNO> WSNAME l-.PASSWDl < OBJNAME 1 ] L0BJNAME2 ...] >
The indicated global objects (but not system variables) are
copied from the indicated workspace (the source workspace)
into the active workspace. The system reports the date and
time at which the source workspace was saved.
If the list of objects to be copied is omitted, all global
objects' other than system variables are copied from the source
workspace.
If the indicated source workspace is for some reason
unavailable, copying cannot take place. The possible errors
in gaining access are the same as those discussed under the
LOAD command, namely IMPROPER LIBRARY REFERENCE, WS NOT FOUND,
WS LOCKED.
When an object to be copied is a group, the group membership
list is copied as well as all those of its members which exist
as global objects in the source workspace.
When an object to be copied has the same name as a global
object in the active workspace, the copied object replaces it.
If there was a shared variable offer with respect to the
variable thus replaced, the offer is retracted.
If names explicitly mentioned in the copy command are not the
names of global objects in the source workspace, the system
reports NOT FOUND: followed by a list of the objects not
found .
Section 8: System Commands 77
Copy During Evaluated Input . A copy command can be entered
while a request for evaluated input (□:) is pending. The
objects copied will then exist in the workspace but do not
become the value of D . The request for evaluated input is
then repeated. However, definition mode may not be entered
during evaluated input, and hence the definition of a function
may not be copied during evaluated input. An attempt to do so
is rejected with the report DEFN ERROR. The following trouble
reports may arise during copying:
WS FULL There is insufficient space to accommodate all the
material to be copied. However, those objects copied before
space was exhausted remain in the active workspace.
SYMBOL TABLE FULL New names occurring in the copied material
exhaust the capacity of the symbol table. Those objects
copied before the symbol table was exhausted remain in the
active workspace.
SI DAMAGE A copied object has replaced the definition of a
halted function, so that its halted execution cannot be
resumed. The user should then enter -*■ enough times to clear
the state indicator of indications of damage.
)PCOPY <LIBNO> WSNAME L-.PASSWDl < OBJNAME L0BJNAME2 ...] >
The protecting copy command works in the same way as the
unprotecting copy, described above, except that a global
object having the same name as one already present in the
active workspace is not copied. If any object is not copied
for this reason, the system reports NOT COPIED: followed by a
list of the objects not copied.
-GROUPING TO FACILITATE COPYING OR ERASURE
It is frequently convenient to copy into the active workspace
several related functions and variables, and to erase them when they are
no longer needed. To facilitate such transfers, a group may be defined
by supplying a list of names that are to be copied or erased together.
Since the group's definition will exist in the workspace, it must have a
name distinct from that of any global object in the workspace.
The definition of a group consists of a list of names. It is not
necessary that objects having those names exist in the workspace. When
a copy command or an erase command mentions the name of a group, then
not only the definition of that group but also all global objects whose
names appear in the membership list are copied (or erased, as the case
may be) .
If the membership of a group A includes the name B of another
group, the act of copying A causes the membership list of B to be
copied, but copying does not extend further. That is, the objects
referred to in the membership list of B are not copied. The same
applies to erasure. If an object named in the membership list of a
group does not exist in the workspace, it cannot be copied or erased,
but no report is generated.
)GROUP GRPNAME LGRPNAME1 L0BJNAME1 0BJNAME2 ...]
A group having the name GRPNAME and consisting of the
membership list shown is formed in the active workspace. If
78 APL Language
the name is already in use as the name of a group, the new
membership list (possibly empty) supersedes the old.
If a group is given an empty membership list, it is dispersed .
Dispersal of a group has no effects on its members (whereas
the command ) ERASE GRPNAME applied to a group expunges both
the group definition and the members) .
If the name of the group appears also as one of the members of
the group, the former membership list is merged with the list
provided in the command, thereby appending new names to the
membership list.
The system makes no acknowledgment of a successful grouping.
An attempt to form a group having a name already in use as the
name of a global variable or function in the workspace is
rejected with the report NOT GROUPED, NAME IN USE.
Since the membership list of a group requires space within the
workspace, the attempt to form a group may find insufficient
space in the workspace or in the symbol table, resulting in
the trouble reports WS FULL or SYMBOL TABLE FULL.
COMMANDS THAT MONITOR THE ACTIVE WORKSPACE
The following commands report aspects of the workspace
environment, but produce no change in it.
) SYMBOLS
)FNS U]
Elicits the report IS ... showing the current maximum number
of symbols.
Reports a list of the global functions in the active
workspace, in alphabetical order [starting with the letter
indicated] .
)VARS U]
)GRPS [A]
Reports a list of the global variables in the active
workspace, in alphabetical order [starting with the letter
indicated] .
Reports a list of the groups in the active workspace, in
alphabetical order [starting with the letter indicated] .
)GRP GRPNAME
Reports the membership of the group named GRPNAME.
)SI
Displays the state indicator, showing the status of halted
functions, with the most recently halted first. The list
Section 8: System Commands 79
shows the name of the function and the number of the statement
at which work is halted. The actions that a user can take
with respect to a halted function are described in Section 7.
Suspended functions are marked in the state indicator by an
asterisk, while pendent functions appear on the state
indicator list without an asterisk. The names of functions
whose definitions have been damaged (by editing, copying, or
erasing them while they are halted) are replaced by blanks in
the state indicator.
)SIV
Displays the state indicator in the same way as )SI , but in
addition, with each function listed, lists names that are
local to its execution.
COMMANDS FOR WORKSPACE STORAGE AND RETRIEVAL
The user may request that a duplicate of the currently active
workspace be saved for later use. When a duplicate of a saved workspace
is subsequently re-activated, the entire environment of computation is
restored as it was, except that variables which were shared in the
active workspace are not automatically shared again when the workspace
is reactivated.
LIBRARIES OF SAVED WORKSPACES
The set of workspaces saved for a particular user is called that
user's library . Each workspace is identified by the user's account
number and the name the user assigns to it. However, in referring to
workspaces in one's own library, the account number may be omitted; the
user's own number is supplied automatically.
In systems with multiple users, it is often convenient to use
functions or variables contributed by others. One user may activate an
entire workspace saved by another user, or may copy from it selected
items. To do so both the library number and the name of the desired
workspace must be supplied. However, the system provides no way of
learning either the account numbers or the names of workspaces belonging
to other users. Thus a user may make use of material from the libraries
of others only if they supply that information. In no case may a user
add, change, or delete material from the library of another user.
Certain libraries (usually identified by a particular group of
library numbers) are not assigned to individual users, but are
designated as public libraries. Any user may obtain a list of
workspaces in a public library, and may use public workspaces. However,
there may be restrictions on who can save, drop, or modify a workspace
in a public library. In general, a public workspace can be re~saved or
deleted only by the user who first saved it.
NAMES AND PASSWORDS FOR WORKSPACES
A saved workspace must be named. The name of a workspace may
duplicate a name used for an APL object within the workspace. A
password may be paired with the name of a workspace. Once this is done,
any reference to the saved workspace (other than dropping it) must be
accompanied by the password.
80 APL Language
The rules governing what workspace names are permissible and what
passwords are permissible may differ from each other and from the rules
governing names within a workspace. In particular, the number of
significant characters in a workspace name or in a password may differ,
and may be less than the number of significant characters in a name used
within a workspace. Workspace names and passwords may be composed of
alphabetic and numeric characters, but not spaces or special symbols;
workspace names must begin with an alphabetic, but this restriction does
not apply to passwords.
)WSID <LIBNO> WSNAME l-.NEWPASS]
Assigns to the active workspace the name indicated, and
(optionally) the library number or the password indicated.
Use of the colon with nothing following it assigns an empty
password; that is, it removes a former password if there was
one. If the active workspace is subsequently saved, future
use of the saved workspace will require use of the password
set here.
Setting of the active workspace's identif cation is
acknowledged by the report WAS . . . followed by the former
name, but not the former password.
)SAVE < <LIBNO> WSNAME <:NEWPASS> >
A duplicate of the active workspace is saved (optionally, in
the indicated public library, otherwise in the user's own
library) under the indicated name, and (optionally) with the
new password indicated. If the user number, workspace name,
or password are omitted, they are supplied from the workspace
identification. After saving, the active workspace has the
same indentif ication (including account number, name, and
password) as the saved workspace.
Although saving does not affect the state of sharing in the
active workspace, current values of the shared variables are
saved in the stored copy.
Saving is acknowledged by a report showing the date and time
at which the workspace was saved.
The command to save the active workspace may be rejected, with
trouble reports as follows:
IMPROPER LIBRARY REFERENCE The system does not permit a
workspace to be saved in the private library of another user,
or in a non-existent public- library, nor does it permit a
workspace named CONTINUE to be saved in a public library.
NOT SAVED, THIS WS IS ... Saving is not permitted when the
name given in the command matches the identification of an
existing saved workspace but does not match the identification
of the active workspace. This restriction prevents the user
from inadvertently overwriting one workspace with another.
WS QUOTA USED UP Saving is permitted only while the number of
workspaces currently saved or the space used by the saved
workspaces is less than the user's allocation. Quotas are set
by the system operator.
WS FULL The workspace contains a shared variable whose value,
when brought in to the workspace, would require for its
storage more work area than is available in the workspace.
Section 8: System Commands 81
NO SPACE There is insufficient space in the system's
auxiliary storage to accomodate the workspace.
LIBRARY TABLE FULL The system's directory of saved workspaces
lacks space in which to record this one.
The last two problems cannot be remedied by the user's action
but require intervention from the system manager to expand the
system facilities.
) CONTINUE WOLD] <:NEWPASS>
The active workspace is saved under the name CONTINUE , and
(unless a different password is specified) with the password
previously assigned to the active workspace. Then the
terminal is signed off in the same way as with the command
)OFF, described below. Unless the CONTINUE workspace is
password-protected, it will be automatically loaded at the
next sign-on.
The system operator has the power to force a sign off
("bounce") some or all users. If a user is bounced, the
system behaves as though the user had entered the command
) CONTINUE. A ) CONTINUE is also executed automatically if the
user's connection to the system is broken without a normal
sign-off; see the discussion of "Automatic Saving After Line
Drop," below.
The command )CONTINUE may be rejected for the same reasons
that apply to )SAVE , listed above.
)L0AD <LIBNO> WSNAME L-.PASSWD]
A duplicate of the indicated workspace (including its entire
computing environment) becomes the user's active workspace.
Shared variable offers in the former active workspace are
retracted. Following a successful )LOAD, the system reports
the date and time at which the loaded workspace was last
saved. The system then immediately executes the latent
expression {OLX) .
Invalid requests to load a workspace may result in the
following trouble reports:
WS NOT FOUND The indicated workspace cannot be found.
WS LOCKED The password supplied in the command does not match
the password of the saved workspace, or is missing from the
command when required.
IMPROPER LIBRARY REFERENCE The user is ineligible to use the
indicated library.
)DROP <LIBNO> WSNAME
The named workspace is removed from the user's library or from
the indicated public library. The system reports the date and
time. _ The password is not required to drop a saved workspace.
Dropping a workspace has no effect on the active workspace.
82 APL Language
An attempt to drop a workspace saved by someone other than the
user is rejected with the report IMPROPER LIBRARY REFERENCE .
Reference to a non-existent workspace is rejected with the
report WS NOT FOUND-
AUTOMATIC SAVING AFTER LINE-DROP
AND SPECIAL PROPERTIES OF THE CONTINUE WORKSPACE
If the connection to the computer or host system is broken before
a normal sign off, and the active workspace is not empty, the system
acts as though the user had entered the command )CONTINUE , thereby
saving the contents of the active workspace under the name CONTINUE -
The CONTINUE workspace is intended solely for temporary storage.
Regardless of the name of the active workspace, it can under these
conditions be saved under the name CONTINUE- That is, a previously
saved workspace named CONTINUE may be replaced by the active workspace
without the protection against overwriting provided for other
workspaces.
Since any account may have a workspace named CONTINUE, to assure
privacy the CONTINUE workspace of one user can not be loaded by another
user, nor can it be saved in a public library. The workspace CONTINUE
does not count as part of the user's quota of saved workspaces.
COMMANDS REGARDING WORKSPACE STORAGE AND RETRIEVAL
)WSID
Reports the identification of the active workspace, showing
the library number if other than the user's own, and the
workspace name, but not the password.
)LIB <LIBNO> [4]
Displays the names of the workspaces in the user's private
library or in the indicated public library. In systems in
which the optional letter of the alphabet can be used the
display is alphabetized. If the command is followed by a
letter of the alphabet, the display begins with names starting
with that letter.
An attempt to display the list of workspaces in another user's
private library or in a non-existent public library is
rejected with the report IM-PROPER LIBRARY REFERENCE-
ACCESS TO THE SYSTEM
Each user of the system is assigned by the system manager an
account identification used to identify data storage and charges for use
of the system. The account identification is required in order to sign
on.
Each user is also assigned a quota indicating the maximum _ number
or capacity of saved workspaces, and an interface quota indicating the
maximum number of variables that may be shared simultaneously.
The system manager may also establish a cpu limit for an account,
limiting the amount of central processor time per keyboard entry. _ When
the limit is reached, an interruption is made in the same way as if a
strong interrupt had been signalled from the keyboard.
Section 8: System Commands 83
SIGN ON
Before work can be started, a physical connection to the
computer must first be established. This may require as
little as turning a switch, or may require establishing a link
from a dial-up terminal to a central computer, possibly
passing through intermediary computing systems which are host
to APL, depending on the system employed and the type of
terminal device employed.
Once communication is established, a numerical identification
of the user must be provided to the system. The precise
manner of providing this varies from system to system.
At sign on, either a clear workspace or the CONTINUE
workspace is activated, depending on the condition which
terminated the preceding session, and the system in use.
Successful sign on is acknowledged by a message showing the
port number, the date and time, the user's name, and the
system identification. This may be preceded by a broadcast
message from the system operator. If CONTINUE was activated,
the system reports the date and time at which it was saved.
An attempt to sign on may be rejected, with one of the
following reports:
ALREADY SIGNED ON The terminal is in use, and must first be
signed off (or bounced by the system operator) before a new
sign-on can be accepted.
NUMBER IN USE The indicated account number is in use at
another port. This may arise either because another user is
in fact using the account, or occasionally (perhaps because of
an equipment problem) because the system did not complete
disconnection at an earlier work session.
NUMBER NOT IN SYSTEM This report means either that the
indicated account has not been enrolled, or that the password
supplied does not match the password last set.
NUMBER LOCKED OUT Authorization for use of the account has
been withdrawn. Workspaces saved in this library cannot be
loaded or copied by anyone.
SIGN OFF
)OFF I HOLD] <:NEWPASS>
An accounting report is displayed, showing the port number,
date and time, and user code, followed by the connect time and
central processor time used at the current session and
cumulatively since the last accounting.
If the word HOLD is used, the connection to the computer or
host system is held for a brief period to accommodate sign on
by another user.
If a colon is used, the word following it (or none) will be
the password for subsequent sign ons.
84 APL Language
)CONTINUE IH0LD~] <:NEWPASS>
Because this command has the dual effects of saving the active
workspace and signing the user off, it is described here and
also in the section on workspace storage and retrieval.
The user is signed off in the same manner as with )OFF, but
the active workspace is first saved under the name CONTINUE.
If a password is set, it will apply at subsequent sign-ons.
The command may be rejected if it is impossible to save the
CONTINUE workspace, either because the physical resources of
the system are unable to accomodate it, or because of a full
workspace produced when the value of a shared variable is
transferred to the workspace. See the preceding discussion of
the commands )SAVE and ) CONTINUE.
Section 8: System Commands 85
INDEX
1050 terminal, 51.
absolute value, 20.
access, to APL system, 73.
access control for shared
variable, 52 53 55 57.
access functions, isolation of, as
programming technique, 4.
access state of shared variable,
55.
account identification, 16 50 83.
account information, 51 .
activation, workspace, and
execution of latent expression,
50 82.
activation, workspace, by load
command, 2 50 51 82.
active workspace, 2 15 50 51 73 79
82.
active workspace, system commands
that affect, 73 79 82.
adding statement, to function
definition, 63.
adding to statement, in function
definition, 64.
addition, 12 18 19.
alpha , 12.
alphabetic characters, 10 12.
ALREADY SIGNED ON, 75 84.
alternating sum, 25.
alternating product, 26.
and, logical, 12 18 20.
announcement from system operator,
received at sign on, 84.
application package, workspace
for, 2 6.
arccos, 18 24.
arcosh, 18 24.
arcsin, 18 24.
arctan, 14 18 24.
argument, of a function, 13 66.
arguments, number of. See
valence,
array, as primitive object, 7 14
17.
arsinh, 18.
artanh, 18.
atomic vector, 50 51.
attention key, as interrupt
signal, 10.
automatic start of function
execution upon load, 51.
availability, of name, 49 60.
axis, operator, 7 27 33 37.
B
backspace, 9 10 71.
bar, 12.
bare output, 72.
base, 12.
base null, 12.
base-value, 42.
BCD terminal, 51 .
billing, system for, 2 6.
binomial, 18 19 24 25.
blank, as separator, 13.
blank, in character constants, 13.
body of a function, 7 59.
boolean functions, 17 21.
bounce, 82.
bracket, 12.
branch, control of sequence, 7 9
61 .
canonical representation, 47 48 59
65.
cap, 12.
cap null, 12.
card reader, communication through
shared variables, 52.
caret, to mark point at which
correction was requested, 10.
carrier return, omitted in bare
output, 72.
carrier return, to release entry
statement, 9.
catenate, 30.
catenate, conformability
requirements, 34.
ceiling, 18 21.
chaining, to join arrays. See
catenate, laminate,
character display, see format,
character editing, of statements
in function definition, 64.
CHARACTER ERROR, 11.
character input, 70.
character input , interrupted , 71.
character set, 10 12.
character representation,
function, 47 48 59 65.
character representation, numbers,
44.
characteristics of APL, 4.
characteristic vector, 30 39.
characters, enterable from
terminal , 9 .
circle, 12 19.
circle bar, 12.
circle slope, 12.
circle stile, 12.
circular functions, 18 23.
classification, names, 49.
CLEAR, command, 74.
clear workspace, 50 76.
CLEAR WS, 50 76.
Index
87
close shoe, 12.
colon, 12 62.
column, 27 41 .
combinations, 18 19 24 25.
comma, 12 33.
comment, 62.
communication, with processors
external to APL, 7 52.
comparison tolerance, 20 22 50 51
76.
complement, logical, 20.
complex roots, undefined, 22.
composite character, 10.
compound expression, 14.
compound interest, example, 1.
compress, 30 36.
compute time, 51.
concurrent process, and shared
variables, 52.
conformability, 11 17 28 34.
conjugate, 18. 20.
connect time, 51 .
connection, broken, 82 83.
connection, establishing, 84.
connection, hold, 74 82 84.
constant, 10.
continuation, output line that
exceeds printing width, 72.
CONTINUE, command, 74. 82 83.
CONTINUE, workspace, 80 84.
CONTINUE HOLD, command, 74 80 83.
coordinate, array, 4 27 37.
COPY, command, 71 74 77 78.
copying, of function definition,
59 71 77.
correction of statement being
entered , 9 .
correction, of entry before
release, 9.
cos, 18 23.
cosh, 18 23.
coupling, degree of, 54 57.
cup , 12.
cursor, to indicate position of
next entry, 9.
D
data, names used for, 1.
data transformations, 43.
date, 50 51.
deal, 18 30 39.
decimal point, in display, 45.
decimal point, in entering
numbers , 10.
decode, 30 42.
defined functions, control of
sequence of execution in, 9 61
defined functions, treated same
Wciy as primitives, 7 66.
definition error, 11 71.
DEFN ERROR, 11 71 75 77
degree of coupling, 54.
del, 12.
del stile, 12.
del tilde, 12.
delay, 48 50.
deleting a statement, of function
definition, 63.
deletion, workspace. See drop,
delta, 12.
delta stile, 12.
DESCRIBE, variable used to explain
an application package, 2.
determinant, function for, as
example, 65.
device, used to display computer's
response, 1 .
diagonal, 35.
dieresis, 12.
dimension, see shape or axis,
disk-file, used in conjunction
with APL, 1 .
display, of function definition,
63.
display, of result of computation,
1 9.
distinguished names, 47.
division, 12 18 19.
division by zero, 17.
domain, of scalar functions, 17.
DOMAIN ERROR, 11.
domain error, produced by locked
function, 69.
domino , 12 40.
dot, 12.
double attention, as strong
interrupt, 10.
down arrow , 12.
downstile, character, 10 12.
drop, 30 36.
DROP, command, 74.
dyadic, valence of function, 13.
dyadic and mondaic forms of
primitive functions, 18.
dyadic functions, identity
elements of, 19.
E
e, base of natural logarithm, 22.
editing, of function definition,
59 63 64.
editing, of function header, 63
64.
empty array, 32.
empty array, reduction over, 17
26.
empty branch, 62.
encode, 30 42.
entry, from keyboard, 1 7 9 70 71
environment of computation, 47.
epsilon, 12.
equal, 12 18 19.
ERASE, command, 74 77.
erasure, dynamic, 49.
error reports, for execute, 43.
error report, form of, 10.
88
APL Language
error report, table of, 11.
error report, when evoked, 10.
establishment, of function
definition, 59.
evaluated input, 70 71.
evaluated input, during copy, 78.
evaluated input, interrupted, 71.
even roots, of negative numbers,
undefined, 22.
exclusive or, 21 .
execute, 30 43.
execute, error report, 43.
execution, of defined function,
66.
execution, order of, 14.
execution, sequence of statements
in defined function, 61 66.
exit, branch to, 62.
exit, from function definition
mode , 64.
expand , 30 36 37.
exponentiation, 22.
exponential, 18 23.
exponential form, see scaled form,
expression, 9.
expunge, object from workspace,
48.
extension of scalar function to
array, 17.
factorial, 18 24.
false, number used to represent,
20.
first, selection of by take, 36.
first axis, 37.
fix, function definition, 47 48
59.
floor, 10 18 21 .
FNS, command, 2 74 79.
format, 30 ; ii Tj *
function, defined, used in
statement, 1 3 5 66.
function, derivation of term, 13.
function, establishment, 47.
function, represented as character
matrix, 6 47.
function definition mode, 59 63
64.
function display, by canonical
representation, 6 65.
function display, in definition
mode , 6 4.
function establishment, 47 48 59.
function establishment, failure
of, 60.
function tables, generated by
outer product, 29.
functions, list of, 2 74 79.
functions, primitive scalar, 18.
functions, used in sample
application, display of, 4 6.
fuzz, see comparison tolerance.
general logarithm, 18 23.
general offer, of shared variable,
54.
global names, as referents of
system commands and del, 61.
global objects, in erase command,
77.
global use of names, 60.
grade, 30 38 39.
greater than, 12 19.
greatest common divisor, 62 69.
grid, of consecutive values, 38.
group, appending new members, 79.
GROUP, command, 74 78.
group, dispersal of, 79.
group, formation of, 78.
group, in erase command, 77.
GRP , command , 74 79.
GRPS, command, 74 79.
H
half duplex, shared variable
access control, 57.
halted execution, of defined
function, 60 66 67.
header, of a function, 7 59 60.
hierarchy, absence of, 14.
HOLD, as part of CONTINUE or HOLD
commands, 74 82 84.
host system, 47.
hyperbolic functions, 18 23.
identification, of user account,
16 50 83.
identity elements, 17 19.
implication, 21 .
IMPLICIT ERROR, 11 50.
IMPROPER LIBRARY REFERENCE, 75 82
83.
INCORRECT COMMAND, 73.
INCORRECT SIGN ON, 75.
INDEX ERROR, 11 .
index generator, 30 38.
index of, 30 38.
index origin, 15 22 37 50 51 76.
indexing, of arrays by arrays, 7
30 37.
inhibition, of set or use of
shared variable, 57.
initial value, of shared variable,
54.
inner product, conformability of
arrays , 28 .
inner product, operator, 7 25.
input from terminal, 1 70 71 .
inserting statement, to function
definition, 63.
integer part, 21 .
Index
89
interface quota, 54.
INTERFACE QUOTA EXHAUSTED , 11.
interrupt, from terminal, 9.
INTERRUPT, system response, 11.
interruption, of character input,
71.
interruption, of evaluated input,
71.
interruption, of pending set or
use of a shared variable, 55.
interruption, of system command,
73.
inventory, system for, 2 4 6.
inverse circular functions, 24.
inverse permutation, 38.
invoice, in sample application, 6.
iota, 12 30 38.
italic, font used for capital
letters, 10.
iterative procedures, use of
branching in, 61 .
I-becim, 12.
K
key, see password.
keyboard, typewriter-like, 1 12.
keying time, 51 .
labels, in function definition,
62.
labels, localization of, 62.
large data array, facilitated by
generalized access, 4.
last axis, 37.
last elements, selection of, by
take, 36.
latent expression, 50 51 76.
least squares approximation, in
matrix division, 41 .
left arrow, 9 12.
left identity, 17.
LENGTH ERROR, 11 17.
less than, 12 18 19.
letters of the alphabet, used to
form names , 1 .
LIB, command, 74.
library, collection of workspaces,
15 16 80.
library, public and private, 16 80
83.
library number, 77.
LIBRARY TABLE FULL, trouble rpeort
during save, 75 82.
line counter, 50 51 76.
line drop, automatic saving of
continue workspace, 82 83.
line zero, of function definition,
62.
list of customers, stored as
matrix, 3.
LOAD, command, 2 15 50 51 73 79
82.
load, workspace, and execution of
latent expression, 51 73 79 82.
local names, in function header,
60.
local use of names, 61.
local variables, and state
indicator, 80.
local variables, in function
execution, 66.
localization, of function
arguments and result, 66.
localization, of system variables,
50.
lock, see password,
locked function, 65 69.
locked function, interruption of,
69.
LOCKED OUT, 84.
log, 12 14.
logarithm, 19 22.
logarithm, absence of general
identity element for, 17.
logical complement, 20.
logical negation, 20.
M
magnitude, 18 20.
material implication, 21.
matrix, 14.
matrix divide, 30 40.
matrix inverse, 30 40.
matrix product, generalized as
inner product, 7 28.
maximum, 18 19 21.
membership, 30 39.
message from system operator,
received at sign on, 84.
minimum , 10 18 19 21.
minus , 18 19.
mixed functions, 17.
mixed functions, classification
of, 29.
monadic, valence of function, 13,
monadic and dyadic forms of
primitive functions, 18.
multiple specification, 9.
multiplication, 12 18 19 25.
multi-dimensional arrays, 14.
N
name, expunged from workspace, 48.
name, standing for variable, 1 9.
name classification, 48 49.
name list, 48 49.
named functions or data, benefits
of using, 2.
names, distinguished, 47.
names , for symbols , 10.
90
APL Language
names, in statements from
keyboard, 1 .
names, rules for forming, 16 80.
nand, 12 18.
natural logarithm, 18 22 23.
negative, 18 20.
negative numbers, sign used for,
13.
new-line signal, omitted from bare
output, 72.
niladic, valence of function, 13.
NO SHARES, system response, 11.
NO SPACE, 82.
nor, 12 18.
not, logical, 18 20.
NOT COPIED, 75 78.
not equal, 12 18 19.
NOT FOUND, 75 77.
not greater than, 12 18 19.
NOT GROUPED, 75 79.
not less than, 12 18 19.
NOT SAVED, 75 81 .
notebook, similiarity to
workspace, 2.
null, 12.
NUMBER IN USE, 75 84.
NUMBER LOCKED OUT, 75 84.
NUMBER NOT IN SYSTEM, 75 84.
numbers, group of used together,
1.
numbers , in statements entered
from keyboard , 1 .
numeric, characters, 10.
numerical functions, 29 40.
OFF, command, 74.
OFF HOLD, command, 74.
offer, shared variable, 50 52 54.
omega , 12.
one-origin indexing, 15.
open shoe, 12.
operators, which modify primitive
functions, 7 13 17 25.
or, logical, 12 18 20.
order, of elements in reshaping an
array, 32.
order of execution, 14.
orders, customer, in sample
application, 4 6.
origin, index, 15 22 37 50 51 76.
out of, combinations, or binomial
coefficients, 18 19 24 25.
outer product, operator, 7 29.
overbar, for negative numbers, 12
13.
overstrike, illegitimate, 11.
overstriking, to form escape
character from character input,
71.
parentheses, 12.
parentheses, used to indicate
order of execution, 14.
password, for user identification,
16 73 84.
password, for workspace, 16 76 77
80 82.
password, in system commands, 73
82.
PCOPY, command, 74 78.
pendent execution, 67.
pendent function, and localization
of names, 61 .
pendent functions, and state
indicator, 67.
permutation, 38.
permutation, random, 39.
pi, 23.
pi times, 18 23.
plus, 12 18 19.
polynomial evaluation, see decode,
power, 18 19 22 25.
precedence, absence of, for
functions, 7 14.
precision, and comparison
tolerance, 20.
precision, of character
representation , 44.
primitive scalar functions, 18.
primitive objects, arrays as, 4
14.
primitive functions,
classification of, 17.
primitive objects, numbers and
symbols as, 1 .
primitive functions, symbols for,
8 10.
printer, used in conjunction with
APL, 1 52.
printing precision, 50 51 72 76.
printing width, 51.
printing width, and bare output,
72.
printing width, in function
definition, 50.
printing width, in normal output,
72 76.
private library, 16.
processor, communicating with APL
via shared variables, 52.
program, sequence of statements, 1
6 61.
projection, on space spanned by
column vectors, obtained from
matrix division, 41.
prompt, by use of bare output, 72.
prompting message, before request
for input, 6.
protecting copy, 74 78.
pseudo-random numbers, 22 39.
public library, 16 80 81 83.
punch cards, use in conjunction
with APL, 52.
Index
91
Pythagorean functions, 18 23,
Q
quad , 12.
quad input, 70 71 .
quad- prime input, 70.
query, 12.
quota, of cpu time per entry, 83.
quota, of saved workspaces, 80 83.
quota, of shared variables, 54 83.
quote, 12.
quote dot, 12.
quote marks, around character
constants, 13.
quote mark, as member of character
vector , 13.
quote quad, 12 70.
reversing half duplex, shared
variable access control, 57.
revision, of entry before release,
9.
revision, of function definition,
59 63 64.
rho, 12.
rho, reshape, 32.
rho, to determine shape of array,
15.
right arrow, 12 61.
right identity, 17.
roll, 18 22.
root, square, etc., 22.
rotate, 30 33.
rotation, 33.
row, 27.
R
radian measure, in argument to
sin, cos, tan, 23.
radix, 42.
random link, 22 50 51 76.
random number generator, 22.
random permutation, 39.
random selection without
replacement, 39.
rank, of an array, 14 32.
rank, of result produced by
indexing, 38.
rank, restrictions on, for
arguments of mixed functions,
31 .
RANK ERROR, 11 .
rank order, 39.
rank vector, 32.
ravel, 30 31.
real numbers, functions on, 17.
reciprocal, 18 20.
recursion, 69.
reduction, operator, 7 25.
reduction over empty array, 17^25.
relations, 18 20.
remainder, 19.
replace statement, in function
definition, 63.
representation, see encode and
format .
RESEND, system response, 11.
reshape, 30 31 .
residue, 18 19.
residue, in definition of encode,
42.
response, to statements entered
from keyboard , 1 .
result, of defined function, 66.
retraction, of shared variable
offer, 57.
return, to release statement being
entered, 9.
reverse, 30 33.
sales, system for, 2 6.
SAVE, command, 74 80.
scalar, as argument of inner
product, 29.
scalar, constant, 10.
scalar, for vector in catenate,
34.
scalar functions, concept, 17.
scalar functions, primitive, list
of, 18.
scalar in place of vector, for
mixed functions, 31.
scaled form, numbers represented
by, 10.
scan, operator, 7 25 26.
selection, functions for, 29 36.
selector generator, 38.
Selectric terminal, 51.
semantic rules , independent of
data representation, 7.
semicolon, 12 37.
sequence of control, 1 7 61.
set membership, 39.
setting shared variable, 52 55.
shadowing, of global names by
local names, 61 .
shape, function, 30 31.
shape, of array, 14 15 32.
shape vector, 32.
shared variable, 50 57.
shared variable offer, 52 53.
shared variable, and workspace
full during save, 80.
shared variable, inquiries
regarding, 53 57.
shared variable, quota, 54.
shared variable, retraction of, 53
57.
shared variable, shared with the
system environment, 50 52.
shared variable, to communicate
with processors outside APL, 7.
SI, command, 11 74 78'.
92 APL Language
SI DAMAGE, 11 68 75 78.
side effects, absence of, 7.
sign off, procedure for, 84.
sign on, procedure for, 84.
significance, of names, 1.
signum, 18 20.
simplex, shared variable access
control, 57.
sin, 14 18 23.
singularity, of matrix to be
inverted, 40.
sinh, 18 23.
SIV, command, 7 4 80.
size, of array, 3 14.
slash, 12.
slash bar, 12.
slope, 12.
slope bar, 12.
space , 12.
spaces, as separators, 13.
special characters, 10.
specification, 9.
specification, multiple, 9.
star, 12.
state indicator, 15 67 74 80.
state indicator damage, 11 68 75
78.
state indicator, clearing, 67 76
79.
state indicator, with display of
local variables, 74 80.
statement, 1 73.
statement, entered from keyboard,
1.
statement, form of, 9.
statement types, listed, 7.
stile, 12.
stop control, 66 68.
storage and retrieval , of
workspaces, 73.
strong interrupt, 9.
structure, functions for, 29.
structure, of arrays, 14.
subtraction, 13 18.
surrogate name, for shared
variable, 52 53.
suspended execution, 61 67 80.
suspended function, and
localization of names, 61 67
80.
suspended function, and state
indicator, 61 67 80.
suspended function, automatic
restart by latent expression,
50.
symbol table, 76 78.
SYMBOL TABLE FULL, 11 75.
SYMBOLS, command, 74 77 79.
symbols, in statements entered
from keyboard , 1 .
symbols, used to represent
primitive functions, 8 10.
syntax, 1 7 14 59 60.
SYNTAX ERROR, 11.
system commands, 16 73 74.
SYSTEM ERROR, 11 .
system functions, 47 48.
system variables, to communicate
with APL environment ,71517
47 50.
table, as matrix, 14.
tables, generated by outer
product, 29.
table-generating functions, 25.
tabular displays, use of format to
generate, 46.
take, 30 36.
tan, 18 23.
tanh, 18 23.
tapes, communication through
shared variable, 1 52.
terminal, common characteristics
of, 9.
terminal, typewriter-like, 9.
terminal, video, 9.
terminal control characters,
obtained from atomic vector,
51.
terminal type, 51.
text processing, 70.
three-dimensional array, example
of, 3.
tilde, 12.
time stamp, 50 51.
times, 12 18 19 25.
top , 12.
top null, 12.
trace Control, 68.
transformations, numeric to
character and vice versa, 29.
transpose, general, 30 35.
transpose, ordinary, 30 35.
trigonometric functions , see
circular functions,
trouble reports, 75.
true, number 1 used to represent,
20.
type, data, 16 29 43.
type element, non-APL, used for
application, 3.'
U
undefined values from certain
expressions, 17.
underbar , 12.
up arrow, 12.
upstile, 12.
use of shared variable facility,
52.
user identification, 16 50 83.
user load, 50 51 .
using shared variable, restriction
imposed by access control, 55.
Index
93
V
valence, function, 7 13.
VALUE ERROR, 1 1 .
variable, named collection of
data, 1 .
variables, identified by name
class function, 49.
variables, list of, 2 74 79.
VARS, command, 2 74 79.
vector , entry of , 10 13 14.
W
weak interrupt, 9.
width, of character
representation, 44.
work area available, 50 51 76.
workspace , active ,2 15.
workspace, as unit of storage, 15
16.
workspace, prepared, as example,
2.
workspace full, 11 57 69 75 78 79
30.
workspace name, 73 74 80.
WS FULL, 11 57 69 75 78 79 80.
WS LOCKED, 75 77 79 82.
WS NOT FOUND, 75 77 79 82.
WSID, command, 73 74 80.
zero divided by zero, defined, 19,
zero-origin indexing, 15.
94 APL Language
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